Rules for integrands of the form(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)
when b c-a d≠0 ∧ b e-a f≠0 ∧ d e-c f≠0

1:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  b c+a d==0 ∧ n==m ∧ m∈ℤ

Derivation: Algebraic simplification

Basis: If  b c+a d==0 ∧ m∈ℤ, then (((a+b x))^m)(((c+d x))^m)==(((a c+b d (x^2)))^m)

Rule 1.1.1.3.1: If  b c+a d==0 ∧ n==m ∧ m∈ℤ, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶   ∫(((a c+b d (x^2)))^m)(((e+f x))^p)dx

Program code:

Int[(a_+b_.*x_)^m_.*(c_+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  Int[(a*c+b*d*x^2)^m*(e+f*x)^p,x] /;
FreeQ[{a,b,c,d,e,f,m,n,p},x] && EqQ[b*c+a*d,0] && EqQ[n,m] && IntegerQ[m] && (NeQ[m,-1] || EqQ[e,0] && (EqQ[p,1] || Not[IntegerQ[p]]))

2.  ∫(a+b x)(((c+d x))^n) (((e+f x))^p)dx

1:  ∫(a+b x)(((c+d x))^n) (((e+f x))^p)dx  when n+p+2≠0 ∧ a d f (n+p+2)-b (d e (n+1)+c f (p+1))==0

Derivation: Quadratic recurrence 2b with c=0 : linear recurrence 2 with a d f (n+p+2)-b (d e (n+1)+c f (p+1))==0

Rule 1.1.1.3.2.1: If  n+p+2≠0 ∧ a d f (n+p+2)-b (d e (n+1)+c f (p+1))==0, then

∫(a+b x)(((c+d x))^n) (((e+f x))^p)dx  ⟶  ((b (((c+d x))^(n+1))(((e+f x))^(p+1)))/(d f(n+p+2)))

Program code:

Int[(a_.+b_.*x_)*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  b*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(d*f*(n+p+2)) /;
FreeQ[{a,b,c,d,e,f,n,p},x] && NeQ[n+p+2,0] && EqQ[a*d*f*(n+p+2)-b*(d*e*(n+1)+c*f*(p+1)),0]

2:  ∫(a+b x)(((c+d x))^n) (((e+f x))^p)dx  when  b c-a d≠0 ∧ ((n|p)∈(SuperMinus[ℤ]) ∨ p==1 ∨ p∈(SuperPlus[ℤ])∧(n∉ℤ ∨ 9p+5(n+2)≤0 ∨ n+p+1≥0))

Derivation: Algebraic expansion

Rule 1.1.1.3.2.2: If  b c-a d≠0 ∧ ((n|p)∈(SuperMinus[ℤ]) ∨ p==1 ∨ p∈(SuperPlus[ℤ])∧(n∉ℤ ∨ 9p+5(n+2)≤0 ∨ n+p+1≥0)), then

∫(a+b x)(((c+d x))^n) (((e+f x))^p)dx  ⟶  ∫ExpandIntegrand[(a+b x)(((c+d x))^n) (((e+f x))^p),x]dx

Program code:

Int[(a_+b_.*x_)*(d_.*x_)^n_.*(e_+f_.*x_)^p_.,x_Symbol] :=
  Int[ExpandIntegrand[(a+b*x)*(d*x)^n*(e+f*x)^p,x],x] /;
FreeQ[{a,b,d,e,f,n},x] && IGtQ[p,0] && EqQ[b*e+a*f,0] && Not[ILtQ[n+p+2,0] && GtQ[n+2*p,0]]

Int[(a_+b_.*x_)*(d_.*x_)^n_.*(e_+f_.*x_)^p_.,x_Symbol] :=
  Int[ExpandIntegrand[(a+b*x)*(d*x)^n*(e+f*x)^p,x],x] /;
FreeQ[{a,b,d,e,f,n},x] && IGtQ[p,0] && (NeQ[n,-1] || EqQ[p,1]) && NeQ[b*e+a*f,0] &&
  (Not[IntegerQ[n]] || LtQ[9*p+5*n,0] || GeQ[n+p+1,0] || GeQ[n+p+2,0] && RationalQ[a,b,d,e,f]) && (NeQ[n+p+3,0] || EqQ[p,1])

Int[(a_.+b_.*x_)*(c_+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  Int[ExpandIntegrand[(a+b*x)*(c+d*x)^n*(e+f*x)^p,x],x] /;
FreeQ[{a,b,c,d,e,f,n},x] && NeQ[b*c-a*d,0] && 
  (ILtQ[n,0] && ILtQ[p,0] || EqQ[p,1] || 
    IGtQ[p,0] && (Not[IntegerQ[n]] || LeQ[9*p+5*(n+2),0] || GeQ[n+p+1,0] || GeQ[n+p+2,0] && RationalQ[a,b,c,d,e,f]))

3:  ∫(a+b x)(((c+d x))^n) (((e+f x))^p)dx  when p<-1 ∧ (n≮-1∨ p∈ℤ)

Derivation: Quadratic recurrence 2b with c=0

Derivation: Quadratic recurrence 3b with c=0,  n=p and p=n

Note: If  n and p are both negative and one is an integer, best to drive that integer exponent toward -1 since the terms of the antiderivative of ((((a+b x))^m)/(c+d x)) are of the form g (((a+b x))^k).

Rule 1.1.1.3.2.3: If  p<-1 ∧ (n≮-1∨ p∈ℤ), then

∫(a+b x)(((c+d x))^n) (((e+f x))^p)dx  ⟶  
-(((b e-a f)(((c+d x))^(n+1))(((e+f x))^(p+1)))/(f(p+1)(c f-d e)))-((a d f(n+p+2)-b (d e(n+1)+c f(p+1)))/(f(p+1)(c f-d e)))∫(((c+d x))^n) (((e+f x))^(p+1))dx

Program code:

Int[(a_.+b_.*x_)*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  -(b*e-a*f)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(f*(p+1)*(c*f-d*e)) - 
  (a*d*f*(n+p+2)-b*(d*e*(n+1)+c*f*(p+1)))/(f*(p+1)*(c*f-d*e))*Int[(c+d*x)^n*(e+f*x)^(p+1),x] /;
FreeQ[{a,b,c,d,e,f,n},x] && LtQ[p,-1] && 
  (Not[LtQ[n,-1]] || IntegerQ[p] || Not[IntegerQ[n] || Not[EqQ[e,0] || Not[EqQ[c,0] || LtQ[p,n]]]])

Int[(a_.+b_.*x_)*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  -(b*e-a*f)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(f*(p+1)*(c*f-d*e)) - 
  (a*d*f*(n+p+2)-b*(d*e*(n+1)+c*f*(p+1)))/(f*(p+1)*(c*f-d*e))*Int[(c+d*x)^n*(e+f*x)^Simplify[p+1],x] /;
FreeQ[{a,b,c,d,e,f,n,p},x] && Not[RationalQ[p]] && SumSimplerQ[p,1]

4:  ∫(a+b x)(((c+d x))^n) (((e+f x))^p)dx  when n+p+2≠0

Derivation: Quadratic recurrence 2b with c=0 : linear recurrence 2

Rule 1.1.1.3.2.4: If  n+p+2≠0, then

∫(a+b x)(((c+d x))^n) (((e+f x))^p)dx  ⟶  
((b (((c+d x))^(n+1))(((e+f x))^(p+1)))/(d f(n+p+2)))+((a d f(n+p+2)-b (d e(n+1)+c f(p+1)))/(d f(n+p+2)))∫(((c+d x))^n) (((e+f x))^p)dx

Program code:

Int[(a_.+b_.*x_)*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  b*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(d*f*(n+p+2)) + 
  (a*d*f*(n+p+2)-b*(d*e*(n+1)+c*f*(p+1)))/(d*f*(n+p+2))*Int[(c+d*x)^n*(e+f*x)^p,x] /;
FreeQ[{a,b,c,d,e,f,n,p},x] && NeQ[n+p+2,0]

3:  ∫(((a+b x))^2) (((c+d x))^n) (((e+f x))^p)dx  when  n+p+2≠0 ∧ n+p+3≠0 ∧ 
d f (n+p+2) ((a^2) d f (n+p+3)-b (b c e+a (d e (n+1)+c f (p+1))))-b (d e (n+1)+c f (p+1)) (a d f (n+p+4)-b (d e (n+2)+c f (p+2)))==0

Derivation: Nondegenerate trilinear recurrence 2 with A=a and B=b : quadratic recurrence 2b with c=0 : linear recurrence 2 with a d f (n+p+2)-b (d e (n+1)+c f (p+1))==0

Rule 1.1.1.3.3: If  n+p+2≠0 ∧ n+p+3≠0 ∧ 
d f (n+p+2) ((a^2) d f (n+p+3)-b (b c e+a (d e (n+1)+c f (p+1))))-b (d e (n+1)+c f (p+1)) (a d f (n+p+4)-b (d e (n+2)+c f (p+2)))==0, then

∫(((a+b x))^2) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
((b (((c+d x))^(n+1)) (((e+f x))^(p+1)) (2 a d f (n+p+3)-b (d e (n+2)+c f (p+2))+b d f (n+p+2) x))/((d^2) (f^2) (n+p+2) (n+p+3)))

Program code:

Int[(a_.+b_.*x_)^2*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  b*(c+d*x)^(n+1)*(e+f*x)^(p+1)*(2*a*d*f*(n+p+3)-b*(d*e*(n+2)+c*f*(p+2))+b*d*f*(n+p+2)*x)/(d^2*f^2*(n+p+2)*(n+p+3)) /;
FreeQ[{a,b,c,d,e,f,n,p},x] && NeQ[n+p+2,0] && NeQ[n+p+3,0] && 
  EqQ[d*f*(n+p+2)*(a^2*d*f*(n+p+3)-b*(b*c*e+a*(d*e*(n+1)+c*f*(p+1))))-b*(d*e*(n+1)+c*f*(p+1))*(a*d*f*(n+p+4)-b*(d*e*(n+2)+c*f*(p+2))),0]

4:  ∫(((a+b x))^m) (((c+d x))^n) (((f x))^p)dx  when  b c+a d==0 ∧ m-n==1

Derivation: Algebraic expansion

Note: Integrals of this form can be expressed as the sum of two hypergeometric functions.

Rule 1.1.1.3.4: If  b c+a d==0 ∧ m-n==1, then

∫(((a+b x))^m) (((c+d x))^n) (((f x))^p)dx  ⟶  a ∫(((a+b x))^n) (((c+d x))^n) (((f x))^p)dx+(b/f) ∫(((a+b x))^n) (((c+d x))^n) (((f x))^(p+1))dx

Program code:

Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(f_.*x_)^p_.,x_Symbol] :=
   a*Int[(a+b*x)^n*(c+d*x)^n*(f*x)^p,x] + b/f*Int[(a+b*x)^n*(c+d*x)^n*(f*x)^(p+1),x] /;
FreeQ[{a,b,c,d,f,m,n,p},x] && EqQ[b*c+a*d,0] && EqQ[m-n-1,0] && Not[RationalQ[p]] && Not[IGtQ[m,0]] && NeQ[m+n+p+2,0]

5.  ∫((((e+f x))^p)/((a+b x)(c+d x)))dx

1:  ∫((((e+f x))^p)/((a+b x)(c+d x)))dx  when  p∈ℤ

Derivation: Algebraic expansion

Rule 1.1.1.3.5.1: If  p∈ℤ, then

∫((((e+f x))^p)/((a+b x)(c+d x)))dx  ⟶  ∫ExpandIntegrand[((((e+f x))^p)/((a+b x)(c+d x))),x]dx

Program code:

Int[(e_.+f_.*x_)^p_./((a_.+b_.*x_)*(c_.+d_.*x_)),x_Symbol] :=
  Int[ExpandIntegrand[(e+f*x)^p/((a+b*x)*(c+d*x)),x],x] /;
FreeQ[{a,b,c,d,e,f},x] && IntegerQ[p]

2.  ∫((((e+f x))^p)/((a+b x)(c+d x)))dx  when  p∉ℤ

1.  ∫((((e+f x))^p)/((a+b x)(c+d x)))dx  when  p>0

1:  ∫((((e+f x))^p)/((a+b x)(c+d x)))dx  when  0<p<1

Derivation: Algebraic expansion

Basis: ((e+f x)/((a+b x) (c+d x)))==((b e-a f)/((b c-a d) (a+b x)))-((d e-c f)/((b c-a d) (c+d x)))

Rule 1.1.1.3.5.2.1.1: If  0<p<1, then

∫((((e+f x))^p)/((a+b x)(c+d x)))dx  ⟶   ((b e-a f)/(b c-a d))∫((((e+f x))^(p-1))/(a+b x))dx-((d e-c f)/(b c-a d))∫((((e+f x))^(p-1))/(c+d x))dx

Program code:

Int[(e_.+f_.*x_)^p_/((a_.+b_.*x_)*(c_.+d_.*x_)),x_Symbol] :=
  (b*e-a*f)/(b*c-a*d)*Int[(e+f*x)^(p-1)/(a+b*x),x] - 
  (d*e-c*f)/(b*c-a*d)*Int[(e+f*x)^(p-1)/(c+d*x),x] /;
FreeQ[{a,b,c,d,e,f},x] && LtQ[0,p,1]

2:  ∫((((e+f x))^p)/((a+b x)(c+d x)))dx  when  p>1

Derivation: Nondegenerate trilinear recurrence 2 with A=a and B=b

Rule 1.1.1.3.5.2.1.2: If  p>1, then

∫((((e+f x))^p)/((a+b x)(c+d x)))dx  ⟶  ((f(((e+f x))^(p-1)))/(b d (p-1)))+(1/(b d))∫(((b d (e^2)-a c (f^2)+f (2 b d e-b c f-a d f) x) (((e+f x))^(p-2)))/((a+b x)(c+d x)))dx

Program code:

Int[(e_.+f_.*x_)^p_/((a_.+b_.*x_)*(c_.+d_.*x_)),x_Symbol] :=
  f*(e+f*x)^(p-1)/(b*d*(p-1)) + 
 1/(b*d)*Int[(b*d*e^2-a*c*f^2+f*(2*b*d*e-b*c*f-a*d*f)*x)*(e+f*x)^(p-2)/((a+b*x)*(c+d*x)),x] /;
FreeQ[{a,b,c,d,e,f},x] && GtQ[p,1]

2:  ∫((((e+f x))^p)/((a+b x)(c+d x)))dx  when  p<-1

Derivation: Nondegenerate trilinear recurrence 3 with A=1 and B=0

Rule 1.1.1.3.5.2.2: If  p<-1, then

∫((((e+f x))^p)/((a+b x)(c+d x)))dx  ⟶  
((f(((e+f x))^(p+1)))/((p+1) (b e-a f) (d e-c f)))+(1/((b e-a f) (d e-c f)))∫(((b d e-b c f-a d f-b d f x) (((e+f x))^(p+1)))/((a+b x)(c+d x)))dx

Program code:

Int[(e_.+f_.*x_)^p_/((a_.+b_.*x_)*(c_.+d_.*x_)),x_Symbol] :=
  f*(e+f*x)^(p+1)/((p+1)*(b*e-a*f)*(d*e-c*f)) + 
  1/((b*e-a*f)*(d*e-c*f))*Int[(b*d*e-b*c*f-a*d*f-b*d*f*x)*(e+f*x)^(p+1)/((a+b*x)*(c+d*x)),x] /;
FreeQ[{a,b,c,d,e,f},x] && LtQ[p,-1]

3:  ∫((((e+f x))^p)/((a+b x)(c+d x)))dx  when  p∉ℤ

Derivation: Algebraic expansion

Basis: (1/((a+b x)(c+d x)))==(b/((b c-a d) (a+b x)))-(d/((b c-a d) (c+d x)))

Rule 1.1.1.3.5.2.3: If  p∉ℤ, then

∫((((e+f x))^p)/((a+b x)(c+d x)))dx  ⟶   (b/(b c-a d))∫((((e+f x))^p)/(a+b x))dx-(d/(b c-a d))∫((((e+f x))^p)/(c+d x))dx

Program code:

Int[(e_.+f_.*x_)^p_/((a_.+b_.*x_)*(c_.+d_.*x_)),x_Symbol] :=
  b/(b*c-a*d)*Int[(e+f*x)^p/(a+b*x),x] - 
  d/(b*c-a*d)*Int[(e+f*x)^p/(c+d*x),x] /;
FreeQ[{a,b,c,d,e,f,p},x] && Not[IntegerQ[p]]

6:  ∫(((((c+d x))^n) (((e+f x))^p))/(a+b x))dx  when  n∈(SuperPlus[ℤ])∧ p<-1

Derivation: Algebraic expansion

Rule 1.1.1.3.6: If  n∈(SuperPlus[ℤ])∧ p<-1, then

∫(((((c+d x))^n) (((e+f x))^p))/(a+b x))dx  ⟶  
∫(((e+f x))^(FractionalPart[p]))ExpandIntegrand[(((((c+d x))^n)(((e+f x))^(IntegerPart[p])))/(a+b x)),x]dx

Program code:

Int[(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_/(a_.+b_.*x_),x_Symbol] :=
  Int[ExpandIntegrand[(e+f*x)^FractionalPart[p],(c+d*x)^n*(e+f*x)^IntegerPart[p]/(a+b*x),x],x] /;
FreeQ[{a,b,c,d,e,f},x] && IGtQ[n,0] && LtQ[p,-1] && FractionQ[p]

7:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  (m|n)∈ℤ ∧ (p∈ℤ ∨ (m>0 ∧ n≥-1))

Derivation: Algebraic expansion

Rule 1.1.1.3.7: If  (m|n)∈ℤ ∧ (p∈ℤ ∨ (m>0 ∧ n≥-1)), then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  ∫ExpandIntegrand[(((a+b x))^m) (((c+d x))^n) (((e+f x))^p),x]dx

Program code:

Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  Int[ExpandIntegrand[(a+b*x)^m*(c+d*x)^n*(e+f*x)^p,x],x] /;
FreeQ[{a,b,c,d,e,f,p},x] && IntegersQ[m,n] && (IntegerQ[p] || GtQ[m,0] && GeQ[n,-1])

8.  ∫(((a+b x))^2)(((c+d x))^n) (((e+f x))^p)dx

1:  ∫(((a+b x))^2) (((c+d x))^n) (((e+f x))^p)dx  when  n<-1

Derivation: ?

Rule 1.1.1.3.8.1: If  n<-1, then

∫(((a+b x))^2) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
(((((b c-a d))^2)(((c+d x))^(n+1))(((e+f x))^(p+1)))/((d^2) (d e-c f) (n+1)))-
(1/((d^2) (d e-c f) (n+1)))∫(((c+d x))^(n+1))(((e+f x))^p)·
((a^2) (d^2) f (n+p+2)+(b^2) c (d e (n+1)+c f (p+1))-2 a b d (d e (n+1)+c f (p+1))-(b^2)d(d e -c f) (n+1) x) dx

Program code:

Int[(a_.+b_.*x_)^2*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  (b*c-a*d)^2*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(d^2*(d*e-c*f)*(n+1)) - 
  1/(d^2*(d*e-c*f)*(n+1))*Int[(c+d*x)^(n+1)*(e+f*x)^p*
    Simp[a^2*d^2*f*(n+p+2)+b^2*c*(d*e*(n+1)+c*f*(p+1))-2*a*b*d*(d*e*(n+1)+c*f*(p+1))-b^2*d*(d*e-c*f)*(n+1)*x,x],x] /;
FreeQ[{a,b,c,d,e,f,n,p},x] && (LtQ[n,-1] || EqQ[n+p+3,0] && NeQ[n,-1] && (SumSimplerQ[n,1] || Not[SumSimplerQ[p,1]]))

2:  ∫(((a+b x))^2) (((c+d x))^n) (((e+f x))^p)dx  when  n+p+3≠0

Derivation: Nondegenerate trilinear recurrence 2 with A=a and B=b

Rule 1.1.1.3.8.2: If  n+p+3≠0, then

∫(((a+b x))^2) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
((b(a+b x)(((c+d x))^(n+1))(((e+f x))^(p+1)))/(d f (n+p+3)))+
(1/(d f (n+p+3)))∫(((c+d x))^n) (((e+f x))^p)·
((a^2) d f (n+p+3)-b (b c e+a (d e (n+1)+c f (p+1)))+b (a d f (n+p+4)-b (d e (n+2)+c f (p+2))) x)dx

Program code:

Int[(a_.+b_.*x_)^2*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  b*(a+b*x)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(d*f*(n+p+3)) + 
  1/(d*f*(n+p+3))*Int[(c+d*x)^n*(e+f*x)^p*
    Simp[a^2*d*f*(n+p+3)-b*(b*c*e+a*(d*e*(n+1)+c*f*(p+1)))+b*(a*d*f*(n+p+4)-b*(d*e*(n+2)+c*f*(p+2)))*x,x],x] /;
FreeQ[{a,b,c,d,e,f,n,p},x] && NeQ[n+p+3,0]

9.  ∫(((((a+b x))^m) (((c+d x))^n))/(e+f x))dx  when  m+n+1==0 ∧ -1<m<0

1:  ∫(1/((((a+b x))^(1/3)) (((c+d x))^(2/3)) (e+f x)))dx

Rule 1.1.1.3.9.1: Let q=(((((d e-c f)/(b e-a f))))^(1/3)) then

∫(1/((((a+b x))^(1/3)) (((c+d x))^(2/3)) (e+f x)))dx  ⟶  
-(((Sqrt[3]) q ArcTan[(1/(Sqrt[3]))+((2 q (((a+b x))^(1/3)))/((Sqrt[3]) (((c+d x))^(1/3))))])/(d e-c f))+((q Log[e+f x])/(2 (d e-c f)))-((3 q Log[q (((a+b x))^(1/3))-(((c+d x))^(1/3))])/(2 (d e-c f)))

Program code:

Int[1/((a_.+b_.*x_)^(1/3)*(c_.+d_.*x_)^(2/3)*(e_.+f_.*x_)),x_Symbol] :=
  With[{q=Rt[(d*e-c*f)/(b*e-a*f),3]},
  -Sqrt[3]*q*ArcTan[1/Sqrt[3]+2*q*(a+b*x)^(1/3)/(Sqrt[3]*(c+d*x)^(1/3))]/(d*e-c*f) + 
  q*Log[e+f*x]/(2*(d*e-c*f)) - 
  3*q*Log[q*(a+b*x)^(1/3)-(c+d*x)^(1/3)]/(2*(d*e-c*f))] /;
FreeQ[{a,b,c,d,e,f},x]

2:  ∫(1/((Sqrt[a+b x]) (Sqrt[c+d x]) (e+f x)))dx  when  2 b d e-f (b c+a d)==0

Derivation: Integration by substitution

Basis: If  2 b d e-f (b c+a d)==0, then (1/((Sqrt[a+b x]) (Sqrt[c+d x]) (e+f x)))==b f Subst[(1/(d (((b e-a f))^2)+b (f^2) (x^2))),x,(Sqrt[a+b x]) (Sqrt[c+d x])](∂_x)((Sqrt[a+b x]) (Sqrt[c+d x]))

Rule 1.1.1.3.9.2: If  2 b d e-f (b c+a d)==0, then

∫(1/((Sqrt[a+b x]) (Sqrt[c+d x]) (e+f x)))dx  ⟶  b f Subst[∫(1/(d (((b e-a f))^2)+b (f^2) (x^2)))dx,x,(Sqrt[a+b x]) (Sqrt[c+d x])]

Program code:

Int[1/(Sqrt[a_.+b_.*x_]*Sqrt[c_.+d_.*x_]*(e_.+f_.*x_)),x_Symbol] :=
  b*f*Subst[Int[1/(d*(b*e-a*f)^2+b*f^2*x^2),x],x,Sqrt[a+b*x]*Sqrt[c+d*x]] /;
FreeQ[{a,b,c,d,e,f},x] && EqQ[2*b*d*e-f*(b*c+a*d),0]

3:  ∫(((((a+b x))^m) (((c+d x))^n))/(e+f x))dx  when  m+n+1==0 ∧ -1<m<0

Derivation: Integration by substitution

Basis: If  m+n+1==0 ∧ -1<m<0, let q=Denominator[m], then (((((a+b x))^m) (((c+d x))^n))/(e+f x))==q Subst[((x^(q (m+1)-1))/(b e-a f-(d e-c f) (x^q))),x,((((a+b x))^(1/q))/(((c+d x))^(1/q)))](∂_x)((((a+b x))^(1/q))/(((c+d x))^(1/q)))

Rule 1.1.1.3.9.3: If  m+n+1==0 ∧ -1<m<0, let q=Denominator[m], then

∫(((((a+b x))^m) (((c+d x))^n))/(e+f x))dx  ⟶  q Subst[∫((x^(q (m+1)-1))/(b e-a f-(d e-c f) (x^q)))dx,x,((((a+b x))^(1/q))/(((c+d x))^(1/q)))]

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_/(e_.+f_.*x_),x_Symbol] :=
  With[{q=Denominator[m]},
  q*Subst[Int[x^(q*(m+1)-1)/(b*e-a*f-(d*e-c*f)*x^q),x],x,(a+b*x)^(1/q)/(c+d*x)^(1/q)]] /;
FreeQ[{a,b,c,d,e,f},x] && EqQ[m+n+1,0] && RationalQ[n] && LtQ[-1,m,0] && SimplerQ[a+b*x,c+d*x]

10:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m+n+p+2==0 ∧ n>0

Derivation: Nondegenerate trilinear recurrence 1 with A=1, B=0 and m+n+p+2==0

Rule 1.1.1.3.10: If  m+n+p+2==0 ∧ n>0, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
(((((a+b x))^(m+1)) (((c+d x))^n) (((e+f x))^(p+1)))/((m+1) (b e-a f)))-((n(d e-c f))/((m+1) (b e-a f)))∫(((a+b x))^(m+1)) (((c+d x))^(n-1))(((e+f x))^p)dx

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  (a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^(p+1)/((m+1)*(b*e-a*f)) - 
  n*(d*e-c*f)/((m+1)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^(n-1)*(e+f*x)^p,x] /;
FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[m+n+p+2,0] && GtQ[n,0] && (SumSimplerQ[m,1] || Not[SumSimplerQ[p,1]]) && NeQ[m,-1]

11.  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m+n+p+3==0

1:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m+n+p+3==0 ∧ a d f (m+1)+b c f (n+1)+b d e (p+1)==0 ∧ m≠-1

Derivation: Nondegenerate trilinear recurrence 3 with A=1 and B=0

Rule 1.1.1.3.11.1: If  m+n+p+3==0 ∧ a d f (m+1)+b c f (n+1)+b d e (p+1)==0 ∧ m≠-1, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  ((b(((a+b x))^(m+1)) (((c+d x))^(n+1))(((e+f x))^(p+1)))/((m+1) (b c-a d) (b e-a f)))

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  b*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) /;
FreeQ[{a,b,c,d,e,f,m,n,p},x] && EqQ[Simplify[m+n+p+3],0] && EqQ[a*d*f*(m+1)+b*c*f*(n+1)+b*d*e*(p+1),0] && NeQ[m,-1]

2:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m+n+p+3==0 ∧ m<-1

Derivation: Nondegenerate trilinear recurrence 3 with A=1 and B=0

Rule 1.1.1.3.11.2: If  m+n+p+3==0 ∧ m<-1, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
((b(((a+b x))^(m+1)) (((c+d x))^(n+1))(((e+f x))^(p+1)))/((m+1) (b c-a d) (b e-a f)))+((a d f (m+1)+b c f (n+1)+b d e (p+1))/((m+1) (b c-a d) (b e-a f)))∫(((a+b x))^(m+1)) (((c+d x))^n) (((e+f x))^p)dx

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  b*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) + 
  (a*d*f*(m+1)+b*c*f*(n+1)+b*d*e*(p+1))/((m+1)*(b*c-a*d)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p,x] /;
FreeQ[{a,b,c,d,e,f,m,n,p},x] && EqQ[Simplify[m+n+p+3],0] && (LtQ[m,-1] || SumSimplerQ[m,1])

12.  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m<-1 ∧ n>0

1:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m<-1 ∧ n>0 ∧ p>0

Derivation: Nondegenerate trilinear recurrence 1 with A=e and B=f

Rule 1.1.1.3.12.1: If  m<-1 ∧ n>0 ∧ p>0, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
(((((a+b x))^(m+1)) (((c+d x))^n) (((e+f x))^p))/(b (m+1)))-(1/(b (m+1)))∫(((a+b x))^(m+1))(((c+d x))^(n-1))(((e+f x))^(p-1))(d e n+c f p+d f(n+p)x)dx

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  (a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p/(b*(m+1)) - 
  1/(b*(m+1))*Int[(a+b*x)^(m+1)*(c+d*x)^(n-1)*(e+f*x)^(p-1)*Simp[d*e*n+c*f*p+d*f*(n+p)*x,x],x] /;
FreeQ[{a,b,c,d,e,f},x] && LtQ[m,-1] && GtQ[n,0] && GtQ[p,0] && (IntegersQ[2*m,2*n,2*p] || IntegersQ[m,n+p] || IntegersQ[p,m+n])

2:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m<-1 ∧ n>1

Derivation: ???

Rule 1.1.1.3.12.2: If  m<-1 ∧ n>1, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
(((b c-a d) (((a+b x))^(m+1)) (((c+d x))^(n-1)) (((e+f x))^(p+1)))/(b (b e-a f) (m+1)))+
(1/(b (b e-a f) (m+1)))∫(((a+b x))^(m+1))(((c+d x))^(n-2))(((e+f x))^p)·
(a d (d e (n-1)+c f (p+1))+b c (d e (m-n+2)-c f (m+p+2))+d (a d f (n+p)+b (d e (m+1)-c f (m+n+p+1)))x)dx

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  (b*c-a*d)*(a+b*x)^(m+1)*(c+d*x)^(n-1)*(e+f*x)^(p+1)/(b*(b*e-a*f)*(m+1)) + 
  1/(b*(b*e-a*f)*(m+1))*Int[(a+b*x)^(m+1)*(c+d*x)^(n-2)*(e+f*x)^p*
    Simp[a*d*(d*e*(n-1)+c*f*(p+1))+b*c*(d*e*(m-n+2)-c*f*(m+p+2))+d*(a*d*f*(n+p)+b*(d*e*(m+1)-c*f*(m+n+p+1)))*x,x],x] /;
FreeQ[{a,b,c,d,e,f,p},x] && LtQ[m,-1] && GtQ[n,1] && (IntegersQ[2*m,2*n,2*p] || IntegersQ[m,n+p] || IntegersQ[p,m+n])

3:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m<-1 ∧ n>0

Derivation: Nondegenerate trilinear recurrence 1 with A=1 and B=0

Rule 1.1.1.3.12.3: If  m<-1 ∧ n>0, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
(((((a+b x))^(m+1)) (((c+d x))^n) (((e+f x))^(p+1)))/((m+1) (b e-a f)))-
(1/((m+1) (b e-a f)))∫(((a+b x))^(m+1)) (((c+d x))^(n-1))(((e+f x))^p)(d e n+c f (m+p+2)+d f (m+n+p+2) x)dx

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  (a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^(p+1)/((m+1)*(b*e-a*f)) - 
  1/((m+1)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^(n-1)*(e+f*x)^p*
    Simp[d*e*n+c*f*(m+p+2)+d*f*(m+n+p+2)*x,x],x] /;
FreeQ[{a,b,c,d,e,f,p},x] && LtQ[m,-1] && GtQ[n,0] && (IntegersQ[2*m,2*n,2*p] || IntegersQ[m,n+p] || IntegersQ[p,m+n])

13:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m>1 ∧ m+n+p+1≠0 ∧ m∈ℤ

Derivation: Nondegenerate trilinear recurrence 2 with A=a and B=b

Note: If the integrand has a positive integer exponent, decrementing it, rather than another positive fractional exponent, produces simpler antiderivatives.

Rule 1.1.1.3.13: If  m>1 ∧ m+n+p+1≠0 ∧ m∈ℤ, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
((b(((a+b x))^(m-1)) (((c+d x))^(n+1))(((e+f x))^(p+1)))/(d f (m+n+p+1)))+
(1/(d f (m+n+p+1)))∫(((a+b x))^(m-2)) (((c+d x))^n) (((e+f x))^p)·
((a^2) d f (m+n+p+1)-b (b c e (m-1)+a (d e (n+1)+c f (p+1)))+b (a d f (2 m+n+p)-b (d e (m+n)+c f (m+p))) x)dx

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  b*(a+b*x)^(m-1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(d*f*(m+n+p+1)) + 
  1/(d*f*(m+n+p+1))*Int[(a+b*x)^(m-2)*(c+d*x)^n*(e+f*x)^p*
    Simp[a^2*d*f*(m+n+p+1)-b*(b*c*e*(m-1)+a*(d*e*(n+1)+c*f*(p+1)))+b*(a*d*f*(2*m+n+p)-b*(d*e*(m+n)+c*f*(m+p)))*x,x],x] /;
FreeQ[{a,b,c,d,e,f,n,p},x] && GtQ[m,1] && NeQ[m+n+p+1,0] && IntegerQ[m]

14:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m>0 ∧ n>0 ∧ m+n+p+1≠0

Derivation: Nondegenerate trilinear recurrence 2 with A=c and B=d

Rule 1.1.1.3.14: If  m>0 ∧ n>0 ∧ m+n+p+1≠0, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
(((((a+b x))^m) (((c+d x))^n) (((e+f x))^(p+1)))/(f (m+n+p+1)))-
(1/(f (m+n+p+1)))∫(((a+b x))^(m-1)) (((c+d x))^(n-1))(((e+f x))^p)(c m(b e-a f)+a n(d e-c f)+(d m(b e-a f)+b n(d e-c f)) x)dx

Program code:

Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  (a+b*x)^m*(c+d*x)^n*(e+f*x)^(p+1)/(f*(m+n+p+1)) - 
  1/(f*(m+n+p+1))*Int[(a+b*x)^(m-1)*(c+d*x)^(n-1)*(e+f*x)^p*
    Simp[c*m*(b*e-a*f)+a*n*(d*e-c*f)+(d*m*(b*e-a*f)+b*n*(d*e-c*f))*x,x],x] /;
FreeQ[{a,b,c,d,e,f,p},x] && GtQ[m,0] && GtQ[n,0] && NeQ[m+n+p+1,0] && (IntegersQ[2*m,2*n,2*p] || (IntegersQ[m,n+p] || IntegersQ[p,m+n]))

15:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m>1 ∧ m+n+p+1≠0

Derivation: Nondegenerate trilinear recurrence 2 with A=a and B=b

Rule 1.1.1.3.15: If  m>1 ∧ m+n+p+1≠0, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
((b(((a+b x))^(m-1)) (((c+d x))^(n+1))(((e+f x))^(p+1)))/(d f (m+n+p+1)))+
(1/(d f (m+n+p+1)))∫(((a+b x))^(m-2)) (((c+d x))^n) (((e+f x))^p)·
((a^2) d f (m+n+p+1)-b (b c e (m-1)+a (d e (n+1)+c f (p+1)))+b (a d f (2 m+n+p)-b (d e (m+n)+c f (m+p))) x)dx

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  b*(a+b*x)^(m-1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/(d*f*(m+n+p+1)) + 
  1/(d*f*(m+n+p+1))*Int[(a+b*x)^(m-2)*(c+d*x)^n*(e+f*x)^p*
    Simp[a^2*d*f*(m+n+p+1)-b*(b*c*e*(m-1)+a*(d*e*(n+1)+c*f*(p+1)))+b*(a*d*f*(2*m+n+p)-b*(d*e*(m+n)+c*f*(m+p)))*x,x],x] /;
FreeQ[{a,b,c,d,e,f,n,p},x] && GtQ[m,1] && NeQ[m+n+p+1,0] && IntegersQ[2*m,2*n,2*p]

16:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m<-1

Derivation: Nondegenerate trilinear recurrence 3 with A=1 and B=0

Note: If the integrand has a negative integer exponent, incrementing it, rather than another negative fractional exponent, produces simpler antiderivatives.

Rule 1.1.1.3.16: If  m<-1, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
((b(((a+b x))^(m+1)) (((c+d x))^(n+1))(((e+f x))^(p+1)))/((m+1) (b c-a d) (b e-a f)))+
(1/((m+1) (b c-a d) (b e-a f)))∫(((a+b x))^(m+1)) (((c+d x))^n) (((e+f x))^p)(a d f (m+1)-b (d e (m+n+2)+c f (m+p+2))-b d f (m+n+p+3) x)dx

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  b*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) + 
  1/((m+1)*(b*c-a*d)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p*
    Simp[a*d*f*(m+1)-b*(d*e*(m+n+2)+c*f*(m+p+2))-b*d*f*(m+n+p+3)*x,x],x] /;
FreeQ[{a,b,c,d,e,f,n,p},x] && ILtQ[m,-1] && (IntegerQ[n] || IntegersQ[2*n,2*p] || ILtQ[m+n+p+3,0])

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  b*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) + 
  1/((m+1)*(b*c-a*d)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p*
    Simp[a*d*f*(m+1)-b*(d*e*(m+n+2)+c*f*(m+p+2))-b*d*f*(m+n+p+3)*x,x],x] /;
FreeQ[{a,b,c,d,e,f,n,p},x] && LtQ[m,-1] && IntegersQ[2*m,2*n,2*p]

17.  ∫((((e+f x))^p)/((a+b x)(Sqrt[c+d x])))dx

1.  ∫(1/((a+b x)(Sqrt[c+d x])(((e+f x))^(1/4))))dx

1:  ∫(1/((a+b x)(Sqrt[c+d x])(((e+f x))^(1/4))))dx  when  -(f/(d e-c f))>0

Derivation: Integration by substitution

Basis: (1/((a+b x) (Sqrt[c+d x]) (((e+f x))^(1/4))))==-4 Subst[((x^2)/((b e-a f-b (x^4)) (Sqrt[c-((d e)/f)+((d (x^4))/f)]))),x,(((e+f x))^(1/4))] (∂_x)(((e+f x))^(1/4))

Rule 1.1.1.3.17.1.1: If  -(f/(d e-c f))>0, then

∫(1/((a+b x)(Sqrt[c+d x])(((e+f x))^(1/4))))dx  ⟶   -4 Subst[∫((x^2)/((b e-a f-b (x^4)) (Sqrt[c-((d e)/f)+((d (x^4))/f)])))dx,x,(((e+f x))^(1/4))]

Program code:

Int[1/((a_.+b_.*x_)*Sqrt[c_.+d_.*x_]*(e_.+f_.*x_)^(1/4)),x_Symbol] :=
  -4*Subst[Int[x^2/((b*e-a*f-b*x^4)*Sqrt[c-d*e/f+d*x^4/f]),x],x,(e+f*x)^(1/4)] /;
FreeQ[{a,b,c,d,e,f},x] && GtQ[-f/(d*e-c*f),0]

2:  ∫(1/((a+b x)(Sqrt[c+d x])(((e+f x))^(1/4))))dx  when  -(f/(d e-c f))≯0

Derivation: Piecewise constant extraction

Basis: (∂_x)((Sqrt[-((f (c+d x))/(d e-c f))])/(Sqrt[c+d x]))==0

Rule 1.1.1.3.17.1.2: If  -(f/(d e-c f))≯0, then

∫(1/((a+b x)(Sqrt[c+d x])(((e+f x))^(1/4))))dx  ⟶   ((Sqrt[-((f (c+d x))/(d e-c f))])/(Sqrt[c+d x])) ∫(1/((a+b x)(Sqrt[-((c f)/(d e-c f))-((d f x)/(d e-c f))])(((e+f x))^(1/4))))dx

Program code:

Int[1/((a_.+b_.*x_)*Sqrt[c_.+d_.*x_]*(e_.+f_.*x_)^(1/4)),x_Symbol] :=
  Sqrt[-f*(c+d*x)/(d*e-c*f)]/Sqrt[c+d*x]*Int[1/((a+b*x)*Sqrt[-c*f/(d*e-c*f)-d*f*x/(d*e-c*f)]*(e+f*x)^(1/4)),x] /;
FreeQ[{a,b,c,d,e,f},x] && Not[GtQ[-f/(d*e-c*f),0]]

2.  ∫(1/((a+b x)(Sqrt[c+d x])(((e+f x))^(3/4))))dx

1:  ∫(1/((a+b x)(Sqrt[c+d x])(((e+f x))^(3/4))))dx  when  -(f/(d e-c f))>0

Derivation: Integration by substitution

Basis: (1/((a+b x) (Sqrt[c+d x]) (((e+f x))^(3/4))))==-4 Subst[(1/((b e-a f-b (x^4)) (Sqrt[c-((d e)/f)+((d (x^4))/f)]))),x,(((e+f x))^(1/4))] (∂_x)(((e+f x))^(1/4))

Rule 1.1.1.3.17.2.1: If  -(f/(d e-c f))>0, then

∫(1/((a+b x) (Sqrt[c+d x]) (((e+f x))^(3/4))))dx  ⟶   -4 Subst[∫(1/((b e-a f-b (x^4)) (Sqrt[c-((d e)/f)+((d (x^4))/f)])))dx,x,(((e+f x))^(1/4))]

Program code:

Int[1/((a_.+b_.*x_)*Sqrt[c_.+d_.*x_]*(e_.+f_.*x_)^(3/4)),x_Symbol] :=
  -4*Subst[Int[1/((b*e-a*f-b*x^4)*Sqrt[c-d*e/f+d*x^4/f]),x],x,(e+f*x)^(1/4)] /;
FreeQ[{a,b,c,d,e,f},x] && GtQ[-f/(d*e-c*f),0]

2:  ∫(1/((a+b x)(Sqrt[c+d x])(((e+f x))^(3/4))))dx  when  -(f/(d e-c f))≯0

Derivation: Piecewise constant extraction

Basis: (∂_x)((Sqrt[-((f (c+d x))/(d e-c f))])/(Sqrt[c+d x]))==0

Rule 1.1.1.3.17.2.2: If  -(f/(d e-c f))≯0, then

∫(1/((a+b x)(Sqrt[c+d x])(((e+f x))^(3/4))))dx  ⟶   ((Sqrt[-((f (c+d x))/(d e-c f))])/(Sqrt[c+d x])) ∫(1/((a+b x)(Sqrt[-((c f)/(d e-c f))-((d f x)/(d e-c f))])(((e+f x))^(3/4))))dx

Program code:

Int[1/((a_.+b_.*x_)*Sqrt[c_.+d_.*x_]*(e_.+f_.*x_)^(3/4)),x_Symbol] :=
  Sqrt[-f*(c+d*x)/(d*e-c*f)]/Sqrt[c+d*x]*Int[1/((a+b*x)*Sqrt[-c*f/(d*e-c*f)-d*f*x/(d*e-c*f)]*(e+f*x)^(3/4)),x] /;
FreeQ[{a,b,c,d,e,f},x] && Not[GtQ[-f/(d*e-c*f),0]]

18.  ∫((Sqrt[e+f x])/((Sqrt[a+b x])(Sqrt[c+d x])))dx

1.  ∫((Sqrt[e+f x])/((Sqrt[b x])(Sqrt[c+d x])))dx  when  d e-c f≠0

1.  ∫((Sqrt[e+f x])/((Sqrt[b x])(Sqrt[c+d x])))dx  when  d e-c f≠0 ∧ c>0 ∧ e>0

1:  ∫((Sqrt[e+f x])/((Sqrt[b x])(Sqrt[c+d x])))dx  when  d e-c f≠0 ∧ c>0 ∧ e>0 ∧ -(b/d)≮0

Rule 1.1.1.3.18.1.1.1: If  d e-c f≠0 ∧ c>0 ∧ e>0 ∧ -(b/d)>0, then

∫((Sqrt[e+f x])/((Sqrt[b x])(Sqrt[c+d x])))dx  ⟶   ((2(Sqrt[e]))/b)(Sqrt[-(b/d)])EllipticE[ArcSin[((Sqrt[b x])/((Sqrt[c])(Sqrt[-(b/d)])))],((c f)/(d e))]

Program code:

Int[Sqrt[e_+f_.*x_]/(Sqrt[b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] :=
  2*Sqrt[e]/b*Rt[-b/d,2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d,2])],c*f/(d*e)] /;
FreeQ[{b,c,d,e,f},x] && NeQ[d*e-c*f,0] && GtQ[c,0] && GtQ[e,0] && Not[LtQ[-b/d,0]]

2:  ∫((Sqrt[e+f x])/((Sqrt[b x])(Sqrt[c+d x])))dx  when  d e-c f≠0 ∧ c>0 ∧ e>0 ∧ -(b/d)<0

Derivation: Piecewise constant extraction

Basis: (∂_x)((Sqrt[-F[x]])/(Sqrt[F[x]]))==0

Rule 1.1.1.3.18.1.1.2: If  d e-c f≠0 ∧ c>0 ∧ e>0 ∧ -(b/d)≯0, then

∫((Sqrt[e+f x])/((Sqrt[b x])(Sqrt[c+d x])))dx  ⟶   ((Sqrt[-b x])/(Sqrt[b x]))∫((Sqrt[e+f x])/((Sqrt[-b x])(Sqrt[c+d x])))dx

Program code:

Int[Sqrt[e_+f_.*x_]/(Sqrt[b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] :=
  Sqrt[-b*x]/Sqrt[b*x]*Int[Sqrt[e+f*x]/(Sqrt[-b*x]*Sqrt[c+d*x]),x] /;
FreeQ[{b,c,d,e,f},x] && NeQ[d*e-c*f,0] && GtQ[c,0] && GtQ[e,0] && LtQ[-b/d,0]

2:  ∫((Sqrt[e+f x])/((Sqrt[b x])(Sqrt[c+d x])))dx  when  d e-c f≠0 ∧ ¬(c>0 ∧ e>0)

Derivation: Piecewise constant extraction

Basis: (∂_x)(((Sqrt[e+f x])(Sqrt[((c+d x)/c)]) )/((Sqrt[c+d x])(Sqrt[((e+f x)/e)])))==0

Rule 1.1.1.3.18.1.2: If  d e-c f≠0 ∧ ¬(c>0 ∧ e>0), then

∫((Sqrt[e+f x])/((Sqrt[b x])(Sqrt[c+d x])))dx  ⟶  (((Sqrt[e+f x])(Sqrt[1+((d x)/c)]) )/((Sqrt[c+d x])(Sqrt[1+((f x)/e)])))∫((Sqrt[1+((f x)/e)])/((Sqrt[b x])(Sqrt[1+((d x)/c)])))dx

Program code:

Int[Sqrt[e_+f_.*x_]/(Sqrt[b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] :=
  Sqrt[e+f*x]*Sqrt[1+d*x/c]/(Sqrt[c+d*x]*Sqrt[1+f*x/e])*Int[Sqrt[1+f*x/e]/(Sqrt[b*x]*Sqrt[1+d*x/c]),x] /;
FreeQ[{b,c,d,e,f},x] && NeQ[d*e-c*f,0] && Not[GtQ[c,0] && GtQ[e,0]]

2.  ∫((Sqrt[e+f x])/((Sqrt[a+b x])(Sqrt[c+d x])))dx

x:  ∫((Sqrt[e+f x])/((Sqrt[a+b x])(Sqrt[c+d x])))dx  when  b e==f (a-1)

Derivation: Algebraic expansion

Basis: If  b e==f (a-1), then ((Sqrt[e+f x])/(Sqrt[a+b x]))==((f (Sqrt[a+b x]))/(b (Sqrt[e+f x])))-(f/(b (Sqrt[a+b x])(Sqrt[e+f x])))

Note: Instead of a single elliptic integral term, this rule produces two simpler such terms.

Rule 1.1.1.3.18.2.x: If  b e==f (a-1), then

∫((Sqrt[e+f x])/((Sqrt[a+b x])(Sqrt[c+d x])))dx  ⟶   (f/b)∫((Sqrt[a+b x])/((Sqrt[c+d x])(Sqrt[e+f x])))dx-(f/b)∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx

Program code:

(* Int[Sqrt[e_.+f_.*x_]/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] :=
  f/b*Int[Sqrt[a+b*x]/(Sqrt[c+d*x]*Sqrt[e+f*x]),x] - 
  f/b*Int[1/(Sqrt[a+b*x]*Sqrt[c+d*x]*Sqrt[e+f*x]),x] /;
FreeQ[{a,b,c,d,e,f},x] && EqQ[b*e-f*(a-1),0] *)

x:  ∫((Sqrt[e+f x])/((Sqrt[a+b x])(Sqrt[c+d x])))dx  when  (b/(b c-a d))>0 ∧ (b/(b e-a f))>0 ∧ -((b c-a d)/d)≮0

Rule 1.1.1.3.18.2.x: If  (b/(b c-a d))>0 ∧ (b/(b e-a f))>0 ∧ -((b c-a d)/d)≮0, then

∫((Sqrt[e+f x])/((Sqrt[a+b x])(Sqrt[c+d x])))dx  ⟶   (2/b)(Sqrt[-((b c-a d)/d)])(Sqrt[((b e-a f)/(b c-a d))])EllipticE[ArcSin[((Sqrt[a+b x])/(Sqrt[-((b c-a d)/d)]))],((f (b c-a d))/(d (b e-a f)))]

Program code:

(* Int[Sqrt[e_.+f_.*x_]/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] :=
  2/b*Rt[-(b*c-a*d)/d,2]*Sqrt[(b*e-a*f)/(b*c-a*d)]*
    EllipticE[ArcSin[Sqrt[a+b*x]/Rt[-(b*c-a*d)/d,2]],f*(b*c-a*d)/(d*(b*e-a*f))] /;
FreeQ[{a,b,c,d,e,f},x] && GtQ[b/(b*c-a*d),0] && GtQ[b/(b*e-a*f),0] && Not[LtQ[-(b*c-a*d)/d,0]] && 
  Not[SimplerQ[c+d*x,a+b*x] && GtQ[-d/(b*c-a*d),0] && GtQ[d/(d*e-c*f),0] && Not[LtQ[(b*c-a*d)/b,0]]] *)

1:  ∫((Sqrt[e+f x])/((Sqrt[a+b x])(Sqrt[c+d x])))dx  when  (b/(b c-a d))>0 ∧ (b/(b e-a f))>0 ∧ -((b c-a d)/d)≮0

Derivation: Integration by substitution

Basis: If  (b/(b c-a d))>0 ∧ (b/(b e-a f))>0, then ((Sqrt[e+f x])/((Sqrt[a+b x])(Sqrt[c+d x])))==((2 (Sqrt[((-b e+a f)/d)]))/(b (Sqrt[-((b c-a d)/d)]))) Subst[((Sqrt[1+((f (x^2))/(b e-a f))])/(Sqrt[1+((d (x^2))/(b c-a d))])),x,(Sqrt[a+b x])] (∂_x)(Sqrt[a+b x])

Basis: ∫((Sqrt[1+((f (x^2))/(b e-a f))])/(Sqrt[1+((d (x^2))/(b c-a d))]))dx==(Sqrt[-((b c-a d)/d)]) EllipticE[ArcSin[(( x)/(Sqrt[-((b c-a d)/d)]))],((f (b c-a d))/(d (b e-a f)))]

Rule 1.1.1.3.18.2.1: If  (b/(b c-a d))>0 ∧ (b/(b e-a f))>0 ∧ -((b c-a d)/d)≮0, then

∫((Sqrt[e+f x])/((Sqrt[a+b x])(Sqrt[c+d x])))dx  ⟶   ((2 (Sqrt[-((b e-a f)/d)]))/(b (Sqrt[-((b c-a d)/d)]))) Subst[∫((Sqrt[1+((f (x^2))/(b e-a f))])/(Sqrt[1+((d (x^2))/(b c-a d))]))dx,x,(Sqrt[a+b x])]
  ⟶   (2/b) (Sqrt[-((b e-a f)/d)]) EllipticE[ArcSin[((Sqrt[a+b x])/(Sqrt[-((b c-a d)/d)]))],((f (b c-a d))/(d (b e-a f)))]

Program code:

Int[Sqrt[e_.+f_.*x_]/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] :=
  2/b*Rt[-(b*e-a*f)/d,2]*EllipticE[ArcSin[Sqrt[a+b*x]/Rt[-(b*c-a*d)/d,2]],f*(b*c-a*d)/(d*(b*e-a*f))] /;
FreeQ[{a,b,c,d,e,f},x] && GtQ[b/(b*c-a*d),0] && GtQ[b/(b*e-a*f),0] && Not[LtQ[-(b*c-a*d)/d,0]] && 
  Not[SimplerQ[c+d*x,a+b*x] && GtQ[-d/(b*c-a*d),0] && GtQ[d/(d*e-c*f),0] && Not[LtQ[(b*c-a*d)/b,0]]]

2:  ∫((Sqrt[e+f x])/((Sqrt[a+b x])(Sqrt[c+d x])))dx  when  ¬((b/(b c-a d))>0 ∧ (b/(b e-a f))>0) ∧ -((b c-a d)/d)≮0

Derivation: Piecewise constant extraction

Basis: (∂_x)(((Sqrt[e+f x])(Sqrt[r (c+d x)]) )/((Sqrt[c+d x])(Sqrt[s (e+f x)])))==0

Note: -((b c-a d)/d)==(-((b c-a d)/d)/.{c->((b c)/(b c-a d)),d->((b d)/(b c-a d)),e->((b e)/(b e-a f)),f->((b f)/(b e-a f))})

Rule 1.1.1.3.18.2.2: If  ¬((b/(b c-a d))>0 ∧ (b/(b e-a f))>0) ∧ -((b c-a d)/d)≮0, then

∫((Sqrt[e+f x])/((Sqrt[a+b x])(Sqrt[c+d x])))dx  ⟶   (((Sqrt[e+f x])(Sqrt[((b (c+d x))/(b c-a d))]) )/((Sqrt[c+d x])(Sqrt[((b (e+f x))/(b e-a f))])))∫((Sqrt[((b e)/(b e-a f))+((b f x)/(b e-a f))])/((Sqrt[a+b x])(Sqrt[((b c)/(b c-a d))+((b d x)/(b c-a d))])))dx

Program code:

Int[Sqrt[e_.+f_.*x_]/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]),x_Symbol] :=
  Sqrt[e+f*x]*Sqrt[b*(c+d*x)/(b*c-a*d)]/(Sqrt[c+d*x]*Sqrt[b*(e+f*x)/(b*e-a*f)])*
    Int[Sqrt[b*e/(b*e-a*f)+b*f*x/(b*e-a*f)]/(Sqrt[a+b*x]*Sqrt[b*c/(b*c-a*d)+b*d*x/(b*c-a*d)]),x] /;
FreeQ[{a,b,c,d,e,f},x] && Not[GtQ[b/(b*c-a*d),0] && GtQ[b/(b*e-a*f),0]] && Not[LtQ[-(b*c-a*d)/d,0]]

19.  ∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx

1.  ∫(1/((Sqrt[b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx

1:  ∫(1/((Sqrt[b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  when  c>0 ∧ e>0

Rule 1.1.1.3.19.1.1: If  c>0 ∧ e>0, then

∫(1/((Sqrt[b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  ⟶  (2/(b(Sqrt[e])))(Sqrt[-(b/d)])EllipticF[ArcSin[((Sqrt[b x])/((Sqrt[c])(Sqrt[-(b/d)])))],((c f)/(d e))]

Program code:

Int[1/(Sqrt[b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] :=
  2/(b*Sqrt[e])*Rt[-b/d,2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d,2])],c*f/(d*e)] /;
FreeQ[{b,c,d,e,f},x] && GtQ[c,0] && GtQ[e,0] && (GtQ[-b/d,0] || LtQ[-b/f,0])

Int[1/(Sqrt[b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] :=
  2/(b*Sqrt[e])*Rt[-b/d,2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d,2])],c*f/(d*e)] /;
FreeQ[{b,c,d,e,f},x] && GtQ[c,0] && GtQ[e,0] && (PosQ[-b/d] || NegQ[-b/f])

2:  ∫(1/((Sqrt[b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  when  ¬(c>0 ∧ e>0)

Derivation: Piecewise constant extraction

Basis: (∂_x)(((Sqrt[1+((d x)/c)]) (Sqrt[1+((f x)/e)]))/((Sqrt[c+d x]) (Sqrt[e+f x])))==0

Rule 1.1.1.3.19.1.2: If  ¬((b/(b c-a d))>0 ∧ (b/(b e-a f))>0), then

∫(1/((Sqrt[b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  ⟶   (((Sqrt[1+((d x)/c)]) (Sqrt[1+((f x)/e)]))/((Sqrt[c+d x]) (Sqrt[e+f x])))∫(1/((Sqrt[b x]) (Sqrt[1+((d x)/c)]) (Sqrt[1+((f x)/e)])))dx

Program code:

Int[1/(Sqrt[b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] :=
  Sqrt[1+d*x/c]*Sqrt[1+f*x/e]/(Sqrt[c+d*x]*Sqrt[e+f*x])*Int[1/(Sqrt[b*x]*Sqrt[1+d*x/c]*Sqrt[1+f*x/e]),x] /;
FreeQ[{b,c,d,e,f},x] && Not[GtQ[c,0] && GtQ[e,0]]

2.  ∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx

1:  ∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  when  (d/b)>0 ∧ (f/b)>0 ∧ c≤((a d)/b) ∧ e≤((a f)/b)

Derivation: Algebraic expansion and integration by substitution

Basis: If  (d/b)>0 ∧ c≤((a d)/b), then (Sqrt[c+d x])==(Sqrt[(d/b)]) (Sqrt[a+b x]) (Sqrt[((b (c+d x))/(d (a+b x)))])

Basis: If  (f/b)>0 ∧ e≤((a f)/b), then (Sqrt[e+f x])==(Sqrt[(f/b)]) (Sqrt[a+b x]) (Sqrt[((b (e+f x))/(f (a+b x)))])

Basis: ((Sqrt[((b (c+d x))/(d (a+b x)))])/((Sqrt[a+b x]) (c+d x) (Sqrt[((b (e+f x))/(f (a+b x)))])))==((2 )/d) Subst[(1/((x^2) (Sqrt[1+((b c-a d)/(d (x^2)))]) (Sqrt[1+((b e-a f)/(f (x^2)))]))),x,(Sqrt[a+b x])] (∂_x)(Sqrt[a+b x])

Basis: ∫(1/((x^2) (Sqrt[1+((b c-a d)/(d (x^2)))]) (Sqrt[1+((b e-a f)/(f (x^2)))])))dx==-(1/(Sqrt[-((b e-a f)/f)])) EllipticF[ArcSin[((Sqrt[-((b e-a f)/f)])/x)],((f (b c-a d))/(d (b e-a f)))]

Rule 1.1.1.3.19.2.1: If  (d/b)>0 ∧ (f/b)>0 ∧ c≤((a d)/b) ∧ e≤((a f)/b), then

∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  ⟶  (Sqrt[(d/f)]) ∫((Sqrt[((b (c+d x))/(d (a+b x)))])/((Sqrt[a+b x]) (c+d x) (Sqrt[((b (e+f x))/(f (a+b x)))])))dx

  ⟶  ((2 (Sqrt[(d/f)]))/d) Subst[∫(1/((x^2) (Sqrt[1+((b c-a d)/(d (x^2)))]) (Sqrt[1+((b e-a f)/(f (x^2)))])))dx,x,(Sqrt[a+b x])]

  ⟶  -((2 (Sqrt[(d/f)]))/(d (Sqrt[-((b e-a f)/f)]))) EllipticF[ArcSin[((Sqrt[-((b e-a f)/f)])/(Sqrt[a+b x]))],((f (b c-a d))/(d (b e-a f)))]

Program code:

Int[1/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] :=
  -2*Sqrt[d/f]/(d*Rt[-(b*e-a*f)/f,2])*EllipticF[ArcSin[Rt[-(b*e-a*f)/f,2]/Sqrt[a+b*x]],f*(b*c-a*d)/(d*(b*e-a*f))] /;
FreeQ[{a,b,c,d,e,f},x] && GtQ[d/b,0] && GtQ[f/b,0] && LeQ[c,a*d/b] && LeQ[e,a*f/b]

x:  ∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  when  -((b e-a f)/f)>0

Derivation: Piecewise constant extraction and integration by substitution

Basis: (∂_x)(((Sqrt[c+d x]) (Sqrt[((b (e+f x))/(f (a+b x)))]))/((Sqrt[e+f x]) (Sqrt[((b (c+d x))/(d (a+b x)))])))==0

Basis: ((Sqrt[((b (c+d x))/(d (a+b x)))])/((Sqrt[a+b x]) (c+d x) (Sqrt[((b (e+f x))/(f (a+b x)))])))==((2 )/d) Subst[(1/((x^2) (Sqrt[1+((b c-a d)/(d (x^2)))]) (Sqrt[1+((b e-a f)/(f (x^2)))]))),x,(Sqrt[a+b x])] (∂_x)(Sqrt[a+b x])

Basis: ∫(1/((x^2) (Sqrt[1+((b c-a d)/(d (x^2)))]) (Sqrt[1+((b e-a f)/(f (x^2)))])))dx==-(1/(Sqrt[-((b e-a f)/f)])) EllipticF[ArcSin[((Sqrt[-((b e-a f)/f)])/x)],((f (b c-a d))/(d (b e-a f)))]

Rule 1.1.1.3.19.2.1: If  -((b e-a f)/f)>0, then

∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  ⟶  (((Sqrt[c+d x]) (Sqrt[((b (e+f x))/(f (a+b x)))]))/((Sqrt[e+f x]) (Sqrt[((b (c+d x))/(d (a+b x)))]))) ∫((Sqrt[((b (c+d x))/(d (a+b x)))])/((Sqrt[a+b x]) (c+d x) (Sqrt[((b (e+f x))/(f (a+b x)))])))dx

  ⟶  ((2 (Sqrt[c+d x]) (Sqrt[((b (e+f x))/(f (a+b x)))]))/(d (Sqrt[e+f x]) (Sqrt[((b (c+d x))/(d (a+b x)))]))) Subst[∫(1/((x^2) (Sqrt[1+((b c-a d)/(d (x^2)))]) (Sqrt[1+((b e-a f)/(f (x^2)))])))dx,x,(Sqrt[a+b x])]

  ⟶  -((2 (Sqrt[c+d x]) (Sqrt[((b (e+f x))/(f (a+b x)))]))/(d (Sqrt[-((b e-a f)/f)]) (Sqrt[e+f x]) (Sqrt[((b (c+d x))/(d (a+b x)))]))) EllipticF[ArcSin[((Sqrt[-((b e-a f)/f)])/(Sqrt[a+b x]))],((f (b c-a d))/(d (b e-a f)))]

Program code:

(* Int[1/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] :=
  -2*Sqrt[c+d*x]*Sqrt[b*(e+f*x)/(f*(a+b*x))]/(d*Rt[-(b*e-a*f)/f,2]*Sqrt[e+f*x]*Sqrt[b*(c+d*x)/(d*(a+b*x))])*
    EllipticF[ArcSin[Rt[-(b*e-a*f)/f,2]/Sqrt[a+b*x]],f*(b*c-a*d)/(d*(b*e-a*f))] /;
FreeQ[{a,b,c,d,e,f},x] && PosQ[-(b*e-a*f)/f] && (* (LtQ[-a/b,-c/d,-e/f] || GtQ[-a/b,-c/d,-e/f]) *)
  Not[SimplerQ[c+d*x,a+b*x] && (PosQ[(-d*e+c*f)/f] || PosQ[(b*e-a*f)/b])] && 
  Not[SimplerQ[e+f*x,a+b*x] && (PosQ[(b*e-a*f)/b] || PosQ[(b*c-a*d)/b])] *)

2:  ∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  when  ((b c-a d)/b)>0 ∧ ((b e-a f)/b)>0 ∧ -(b/d)>0

Derivation: Integration by substitution

Basis: If  ((b c-a d)/b)>0 ∧ ((b e-a f)/b)>0, then (1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))==((2 )/(b (Sqrt[((b c-a d)/b)]) (Sqrt[((b e-a f)/b)]))) Subst[(1/((Sqrt[1+((d (x^2))/(b c-a d))]) (Sqrt[1+((f (x^2))/(b e-a f))]))),x,(Sqrt[a+b x])] (∂_x)(Sqrt[a+b x])

Basis: ∫(1/((Sqrt[1+((d (x^2))/(b c-a d))]) (Sqrt[1+((f (x^2))/(b e-a f))])))dx==(Sqrt[-(b/d)]) (Sqrt[((b c-a d)/b)]) EllipticF[ArcSin[(( x)/((Sqrt[-(b/d)]) (Sqrt[((b c-a d)/b)])))],((f (b c-a d))/(d (b e-a f)))]

Rule 1.1.1.3.19.2.2: If  ((b c-a d)/b)>0 ∧ ((b e-a f)/b)>0 ∧ -(b/d)>0, then

∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  ⟶  ((2 )/(b (Sqrt[((b c-a d)/b)]) (Sqrt[((b e-a f)/b)]))) Subst[∫(1/((Sqrt[1+((d (x^2))/(b c-a d))]) (Sqrt[1+((f (x^2))/(b e-a f))])))dx,x,(Sqrt[a+b x])]
  ⟶   ((2 (Sqrt[-(b/d)]))/(b (Sqrt[((b e-a f)/b)]))) EllipticF[ArcSin[((Sqrt[a+b x])/((Sqrt[-(b/d)]) (Sqrt[((b c-a d)/b)])))],((f (b c-a d))/(d (b e-a f)))]

Program code:

Int[1/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] :=
  2*Rt[-b/d,2]/(b*Sqrt[(b*e-a*f)/b])*EllipticF[ArcSin[Sqrt[a+b*x]/(Rt[-b/d,2]*Sqrt[(b*c-a*d)/b])],f*(b*c-a*d)/(d*(b*e-a*f))] /;
FreeQ[{a,b,c,d,e,f},x] && GtQ[(b*c-a*d)/b,0] && GtQ[(b*e-a*f)/b,0] && PosQ[-b/d] && 
  Not[SimplerQ[c+d*x,a+b*x] && GtQ[(d*e-c*f)/d,0] && GtQ[-d/b,0]] && 
  Not[SimplerQ[c+d*x,a+b*x] && GtQ[(-b*e+a*f)/f,0] && GtQ[-f/b,0]] && 
  Not[SimplerQ[e+f*x,a+b*x] && GtQ[(-d*e+c*f)/f,0] && GtQ[(-b*e+a*f)/f,0] && (PosQ[-f/d] || PosQ[-f/b])]

Int[1/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] :=
  2*Rt[-b/d,2]/(b*Sqrt[(b*e-a*f)/b])*EllipticF[ArcSin[Sqrt[a+b*x]/(Rt[-b/d,2]*Sqrt[(b*c-a*d)/b])],f*(b*c-a*d)/(d*(b*e-a*f))] /;
FreeQ[{a,b,c,d,e,f},x] && GtQ[b/(b*c-a*d),0] && GtQ[b/(b*e-a*f),0] && SimplerQ[a+b*x,c+d*x] && SimplerQ[a+b*x,e+f*x] && 
  (PosQ[-(b*c-a*d)/d] || NegQ[-(b*e-a*f)/f]) (* && PosQ[-b/d] *)

3:  ∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  when  ((b c-a d)/b)≯0

Derivation: Piecewise constant extraction

Basis: (∂_x)((Sqrt[((b (c+d x))/(b c-a d))])/(Sqrt[c+d x]))==0

Rule 1.1.1.3.19.2.3: If  ((b c-a d)/b)≯0, then

∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  ⟶   ((Sqrt[((b (c+d x))/(b c-a d))])/(Sqrt[c+d x]))∫(1/((Sqrt[a+b x])(Sqrt[((b c)/(b c-a d))+((b d x)/(b c-a d))]) (Sqrt[e+f x])))dx

Program code:

Int[1/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] :=
  Sqrt[b*(c+d*x)/(b*c-a*d)]/Sqrt[c+d*x]*Int[1/(Sqrt[a+b*x]*Sqrt[b*c/(b*c-a*d)+b*d*x/(b*c-a*d)]*Sqrt[e+f*x]),x] /;
FreeQ[{a,b,c,d,e,f},x] && Not[GtQ[(b*c-a*d)/b,0]] && SimplerQ[a+b*x,c+d*x] && SimplerQ[a+b*x,e+f*x]

4:  ∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  when  ((b e-a f)/b)≯0

Derivation: Piecewise constant extraction

Basis: (∂_x)((Sqrt[((b (e+f x))/(b e-a f))])/(Sqrt[e+f x]))==0

Rule 1.1.1.3.19.2.4: If  ((b e-a f)/b)≯0, then

∫(1/((Sqrt[a+b x])(Sqrt[c+d x])(Sqrt[e+f x])))dx  ⟶   ((Sqrt[((b (e+f x))/(b e-a f))])/(Sqrt[e+f x]))∫(1/((Sqrt[a+b x])(Sqrt[c+d x]) (Sqrt[((b e)/(b e-a f))+((b f x)/(b e-a f))])))dx

Program code:

Int[1/(Sqrt[a_+b_.*x_]*Sqrt[c_+d_.*x_]*Sqrt[e_+f_.*x_]),x_Symbol] :=
  Sqrt[b*(e+f*x)/(b*e-a*f)]/Sqrt[e+f*x]*Int[1/(Sqrt[a+b*x]*Sqrt[c+d*x]*Sqrt[b*e/(b*e-a*f)+b*f*x/(b*e-a*f)]),x] /;
FreeQ[{a,b,c,d,e,f},x] && Not[GtQ[(b*e-a*f)/b,0]]

20.  ∫((((a+b x))^m)/((((c+d x))^(1/3)) (((e+f x))^(1/3))))dx  when  2 b d e-b c f-a d f==0 ∧ m∈(SuperMinus[ℤ])

1:  ∫(1/((a+b x) (((c+d x))^(1/3)) (((e+f x))^(1/3))))dx  when  2 b d e-b c f-a d f==0

Rule 1.1.1.3.20.1: If  2 b d e-b c f-a d f==0, let q=(((((b (b e-a f))/(((b c-a d))^2))))^(1/3)), then

∫(1/((a+b x) (((c+d x))^(1/3)) (((e+f x))^(1/3))))dx  ⟶  -((Log[a+b x])/(2 q (b c-a d)))-(((Sqrt[3]) ArcTan[(1/(Sqrt[3]))+((2 q (((c+d x))^(2/3)))/((Sqrt[3]) (((e+f x))^(1/3))))])/(2 q (b c-a d)))+((3 Log[q (((c+d x))^(2/3))-(((e+f x))^(1/3))])/(4 q (b c-a d)))

Program code:

Int[1/((a_.+b_.*x_)*(c_.+d_.*x_)^(1/3)*(e_.+f_.*x_)^(1/3)),x_Symbol] :=
  With[{q=Rt[b*(b*e-a*f)/(b*c-a*d)^2,3]},
  -Log[a+b*x]/(2*q*(b*c-a*d)) - 
  Sqrt[3]*ArcTan[1/Sqrt[3]+2*q*(c+d*x)^(2/3)/(Sqrt[3]*(e+f*x)^(1/3))]/(2*q*(b*c-a*d)) + 
  3*Log[q*(c+d*x)^(2/3)-(e+f*x)^(1/3)]/(4*q*(b*c-a*d))] /;
FreeQ[{a,b,c,d,e,f},x] && EqQ[2*b*d*e-b*c*f-a*d*f,0]

2:  ∫((((a+b x))^m)/((((c+d x))^(1/3)) (((e+f x))^(1/3))))dx  when  2 b d e-b c f-a d f==0 ∧ m+1∈(SuperMinus[ℤ])

Derivation: Nondegenerate trilinear recurrence 3 with A=1 and B=0

Rule 1.1.1.3.20.2: If  2 b d e-b c f-a d f==0 ∧ m+1∈(SuperMinus[ℤ]), then

∫((((a+b x))^m)/((((c+d x))^(1/3)) (((e+f x))^(1/3))))dx  ⟶  
((b(((a+b x))^(m+1)) (((c+d x))^(2/3))(((e+f x))^(2/3)))/((m+1) (b c-a d) (b e-a f)))+(f/(6 (m+1) (b c-a d) (b e-a f)))∫(((((a+b x))^(m+1)) (a d (3 m+1)-3 b c (3 m+5)-2 b d (3 m+7) x))/((((c+d x))^(1/3)) (((e+f x))^(1/3))))dx

Program code:

Int[(a_.+b_.*x_)^m_/((c_.+d_.*x_)^(1/3)*(e_.+f_.*x_)^(1/3)),x_Symbol] :=
  b*(a+b*x)^(m+1)*(c+d*x)^(2/3)*(e+f*x)^(2/3)/((m+1)*(b*c-a*d)*(b*e-a*f)) + 
  f/(6*(m+1)*(b*c-a*d)*(b*e-a*f))*
    Int[(a+b*x)^(m+1)*(a*d*(3*m+1)-3*b*c*(3*m+5)-2*b*d*(3*m+7)*x)/((c+d*x)^(1/3)*(e+f*x)^(1/3)),x] /;
FreeQ[{a,b,c,d,e,f},x] && EqQ[2*b*d*e-b*c*f-a*d*f,0] && ILtQ[m,-1]

21.  ∫(((a+b x))^m) (((c+d x))^n)(((f x))^p)dx  when  b c+a d==0 ∧ m-n==0

x:  ∫(((a+b x))^m) (((c+d x))^n)(((f x))^p)dx  when  b c+a d==0 ∧ n==m

Derivation: Piecewise constant extraction

Basis: If  b c+a d==0, then (∂_x)(((((a+b x))^m) (((c+d x))^m))/(((a c+b d (x^2)))^m))==0

Rule 1.1.1.3.21: If  b c+a d==0 ∧ n==m, then

∫(((a+b x))^m) (((c+d x))^n)(((f x))^p)dx  ⟶  (((((a+b x))^m) (((c+d x))^m))/(((a c+b d (x^2)))^m)) ∫(((a c+b d (x^2)))^m)(((f x))^p)dx

Program code:

(* Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(f_.*x_)^p_.,x_Symbol] :=
  Simp[(a+b*x)^m*(c+d*x)^m/(a*c+b*d*x^2)^m]*Int[(a*c+b*d*x^2)^m*(f*x)^p,x] /;
FreeQ[{a,b,c,d,f,m,n,p},x] && EqQ[b*c+a*d,0] && EqQ[n,m] *)

1:  ∫(((a+b x))^m) (((c+d x))^n)(((f x))^p)dx  when  b c+a d==0 ∧ n==m ∧ a>0 ∧ c>0

Derivation: Algebraic simplification

Basis: If  b c+a d==0 ∧ a>0 ∧ c>0, then (((a+b x))^m)(((c+d x))^m)==(((a c+b d (x^2)))^m)

Rule 1.1.1.3.21.1: If  b c+a d==0 ∧ n==m ∧ a>0 ∧ c>0, then

∫(((a+b x))^m) (((c+d x))^n)(((f x))^p)dx  ⟶  ∫(((a c+b d (x^2)))^m)(((f x))^p)dx

Program code:

Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(f_.*x_)^p_.,x_Symbol] :=
  Int[(a*c+b*d*x^2)^m*(f*x)^p,x] /;
FreeQ[{a,b,c,d,f,m,n,p},x] && EqQ[b*c+a*d,0] && EqQ[n,m] && GtQ[a,0] && GtQ[c,0]

2:  ∫(((a+b x))^m) (((c+d x))^n)(((f x))^p)dx  when  b c+a d==0 ∧ n==m

Derivation: Piecewise constant extraction

Basis: If  b c+a d==0, then (∂_x)(((((a+b x))^m) (((c+d x))^m))/(((a c+b d (x^2)))^m))==0

Basis: If  b c+a d==0, then (((((a+b x))^m) (((c+d x))^m))/(((a c+b d (x^2)))^m))==(((((a+b x))^(FracPart[m])) (((c+d x))^(FracPart[m])))/(((a c+b d (x^2)))^(FracPart[m])))

Rule 1.1.1.3.21.2: If  b c+a d==0 ∧ n==m, then

∫(((a+b x))^m) (((c+d x))^n)(((f x))^p)dx  ⟶  (((((a+b x))^(FracPart[m])) (((c+d x))^(FracPart[m])))/(((a c+b d (x^2)))^(FracPart[m])))∫(((a c+b d (x^2)))^m)(((f x))^p)dx

Program code:

Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(f_.*x_)^p_.,x_Symbol] :=
  (a+b*x)^FracPart[m]*(c+d*x)^FracPart[m]/(a*c+b*d*x^2)^FracPart[m]*Int[(a*c+b*d*x^2)^m*(f*x)^p,x] /;
FreeQ[{a,b,c,d,f,m,n,p},x] && EqQ[b*c+a*d,0] && EqQ[n,m]

22:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∈(SuperPlus[ℤ]) ∨ (m|n)∈(SuperMinus[ℤ])

Derivation: Algebraic expansion

Rule 1.1.1.3.22: If  m∈(SuperPlus[ℤ]) ∨ (m|n)∈(SuperMinus[ℤ]), then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  ∫ExpandIntegrand[(((a+b x))^m) (((c+d x))^n) (((e+f x))^p),x]dx

Program code:

Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  Int[ExpandIntegrand[(a+b*x)^m*(c+d*x)^n*(e+f*x)^p,x],x] /;
FreeQ[{a,b,c,d,e,f,n,p},x] && (IGtQ[m,0] || ILtQ[m,0] && ILtQ[n,0])

23:  ∫(((e x))^p) (((a+b x))^m) (((c+d x))^n)dx  when  b c-a d≠0 ∧ p∈𝔽 ∧ m∈ℤ

Derivation: Integration by substitution

Basis: If  k∈(SuperPlus[ℤ]), then (((e x))^p) F[x]==(k/e) Subst[(x^(k (p+1)-1))F[((x^k)/e)],x,(((e x))^(1/k))] (∂_x)(((e x))^(1/k))

Rule 1.1.1.3.23 If  b c-a d≠0 ∧ p∈𝔽 ∧ m∈ℤ, let k=Denominator[p], then

∫(((e x))^p) (((a+b x))^m) (((c+d x))^n)dx  ⟶  (k/e) Subst[∫(x^(k (p+1)-1)) (((a+((b (x^k))/e)))^m) (((c+((d (x^k))/e)))^n) dx,x,(((e x))^(1/k))]

Program code:

Int[(e_.*x_)^p_*(a_+b_.*x_)^m_*(c_+d_.*x_)^n_,x_Symbol] :=
  With[{k=Denominator[p]},
  k/e*Subst[Int[x^(k*(p+1)-1)*(a+b*x^k/e)^m*(c+d*x^k/e)^n,x],x,(e*x)^(1/k)]] /;
FreeQ[{a,b,c,d,e,m,n},x] && NeQ[b*c-a*d,0] && FractionQ[p] && IntegerQ[m]

24.  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m+n+p∈ℤ

1:  ∫(((((a+b x))^m) (((c+d x))^n))/(((e+f x))^2))dx  when  m+n∈(SuperPlus[ℤ])∧ 2 b d e-f (b c+a d)==0

Derivation: Algebraic expansion

Basis: (((((a+b x))^m) (((c+d x))^n))/(((e+f x))^2))==((b d)/(f^2)) (((a+b x))^(m-1)) (((c+d x))^(n-1))+(((b e-a f) (d e-c f) (((a+b x))^(m-1)) (((c+d x))^(n-1)))/((f^2) (((e+f x))^2)))

Rule 1.1.1.3.24.1: If  m+n∈(SuperPlus[ℤ])∧ 2 b d e-f (b c+a d)==0, then

∫(((((a+b x))^m) (((c+d x))^n))/(((e+f x))^2))dx  ⟶  ((b d)/(f^2)) ∫(((a+b x))^(m-1)) (((c+d x))^(n-1))dx+(((b e-a f) (d e-c f))/(f^2)) ∫(((((a+b x))^(m-1)) (((c+d x))^(n-1)))/(((e+f x))^2))dx

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_/(e_.+f_.*x_)^2,x_Symbol] :=
  b*d/f^2*Int[(a+b*x)^(m-1)*(c+d*x)^(n-1),x] + 
  (b*e-a*f)*(d*e-c*f)/f^2*Int[(a+b*x)^(m-1)*(c+d*x)^(n-1)/(e+f*x)^2,x] /;
FreeQ[{a,b,c,d,e,f,m,n},x] && IGtQ[m+n,0] && EqQ[2*b*d*e-f*(b*c+a*d),0]

2:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m+n+p==0 ∧ p∈(SuperMinus[ℤ])

Derivation: Algebraic expansion

Basis: If  m+n+p==0, then (((a+b x))^m) (((c+d x))^n) (((e+f x))^p)==(((f^(p-1)) (((a+b x))^m) (d e p-c f (p-1)+d f x))/((d^p) (((c+d x))^(m+1))))+(((f^(p-1)) (((a+b x))^m) (((e+f x))^p))/(((c+d x))^(m+1))) ((f^(-p+1)) (((c+d x))^(-p+1))-(d^(-p)) (d e p-c f (p-1)+d f x) (((e+f x))^(-p)))

Note: If  p∈(SuperMinus[ℤ]), then (f^(-p+1)) (((c+d x))^(-p+1))-(d^(-p)) (d e p-c f (p-1)+d f x) (((e+f x))^(-p)) is a polynomial of degree -p-1 in x.

Rule 1.1.1.3.24.2: If  m+n+p==0 ∧ p∈(SuperMinus[ℤ]), then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
((f^(p-1))/(d^p)) ∫(((((a+b x))^m) (d e p-c f (p-1)+d f x))/(((c+d x))^(m+1)))dx+(f^(p-1)) ∫(((((a+b x))^m) (((e+f x))^p))/(((c+d x))^(m+1))) ((f^(-p+1)) (((c+d x))^(-p+1))-(d^(-p)) (d e p-c f (p-1)+d f x) (((e+f x))^(-p)))dx

Program code:

Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_,x_Symbol] :=
  f^(p-1)/d^p*Int[(a+b*x)^m*(d*e*p-c*f*(p-1)+d*f*x)/(c+d*x)^(m+1),x] + 
  f^(p-1)*Int[(a+b*x)^m*(e+f*x)^p/(c+d*x)^(m+1)*
    ExpandToSum[f^(-p+1)*(c+d*x)^(-p+1)-(d*e*p-c*f*(p-1)+d*f*x)/(d^p*(e+f*x)^p),x],x] /;
FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[m+n+p,0] && ILtQ[p,0] && (LtQ[m,0] || SumSimplerQ[m,1] || Not[LtQ[n,0] || SumSimplerQ[n,1]])

3:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m+n+p+1==0 ∧ p∈(SuperMinus[ℤ])

Derivation: Algebraic expansion

Basis: If  m+n+p+1==0, then (((a+b x))^m) (((c+d x))^n) (((e+f x))^p)==((b (d^(m+n)) (f^p) (((a+b x))^(m-1)))/(((c+d x))^m))+(((((a+b x))^(m-1)) (((e+f x))^p))/(((c+d x))^m)) ((a+b x) (((c+d x))^(-p-1))-b (d^(-p-1)) (f^p) (((e+f x))^(-p)))

Note: If  p∈(SuperMinus[ℤ]), then (a+b x) (((c+d x))^(-p-1))-b (d^(-p-1)) (f^p) (((e+f x))^(-p)) is a polynomial of degree -p-1 in x.

Rule 1.1.1.3.24.3: If  m+n+p+1==0 ∧ p∈(SuperMinus[ℤ]), then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  b (d^(m+n)) (f^p) ∫((((a+b x))^(m-1))/(((c+d x))^m))dx+∫(((((a+b x))^(m-1)) (((e+f x))^p))/(((c+d x))^m)) ((a+b x) (((c+d x))^(-p-1))-b (d^(-p-1)) (f^p) (((e+f x))^(-p)))dx

Program code:

Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_,x_Symbol] :=
  b*d^(m+n)*f^p*Int[(a+b*x)^(m-1)/(c+d*x)^m,x] + 
  Int[(a+b*x)^(m-1)*(e+f*x)^p/(c+d*x)^m*ExpandToSum[(a+b*x)*(c+d*x)^(-p-1)-(b*d^(-p-1)*f^p)/(e+f*x)^p,x],x] /;
FreeQ[{a,b,c,d,e,f,m,n},x] && EqQ[m+n+p+1,0] && ILtQ[p,0] && (GtQ[m,0] || SumSimplerQ[m,-1] || Not[GtQ[n,0] || SumSimplerQ[n,-1]])

4.  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m+n+p+2==0

1:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m+n+p+2==0 ∧ n∈(SuperMinus[ℤ])

Rule 1.1.1.3.24.4.1: If  m+n+p+2==0 ∧ n∈(SuperMinus[ℤ]), then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
(((((b c-a d))^n) (((a+b x))^(m+1)))/((m+1) (((b e-a f))^(n+1)) (((e+f x))^(m+1))))Hypergeometric2F1[m+1,-n, m+2,-(((d e-c f) (a+b x))/((b c-a d) (e+f x)))]

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_,x_Symbol] :=
  (b*c-a*d)^n*(a+b*x)^(m+1)/((m+1)*(b*e-a*f)^(n+1)*(e+f*x)^(m+1))*
    Hypergeometric2F1[m+1,-n,m+2,-(d*e-c*f)*(a+b*x)/((b*c-a*d)*(e+f*x))] /;
FreeQ[{a,b,c,d,e,f,m,p},x] && EqQ[m+n+p+2,0] && ILtQ[n,0] && (SumSimplerQ[m,1] || Not[SumSimplerQ[p,1]]) && Not[ILtQ[m,0]]

2:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m+n+p+2==0 ∧ n∉ℤ

Derivation: Piecewise constant extraction

Basis: (∂_x)(((((c+d x))^n)/(((e+f x))^n)) ((((((b e-a f) (c+d x))/((b c-a d) (e+f x)))))^(-n)))==0

Rule 1.1.1.3.24.4.2: If  m+n+p+2==0 ∧ n∉ℤ, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx

  ⟶  ((((c+d x))^n)/(((e+f x))^n)) ((((((b e-a f) (c+d x))/((b c-a d) (e+f x)))))^(-n)) ∫((((a+b x))^m)/(((e+f x))^(m+2))) ((((((b e-a f) (c+d x))/((b c-a d) (e+f x)))))^n)dx

  ⟶  (((((a+b x))^(m+1)) (((c+d x))^n) (((e+f x))^(p+1)))/((b e-a f) (m+1)))((((((b e-a f) (c+d x))/((b c-a d) (e+f x)))))^(-n))Hypergeometric2F1[m+1,-n, m+2,-(((d e-c f) (a+b x))/((b c-a d) (e+f x)))]

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] :=
  (a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^(p+1)/((b*e-a*f)*(m+1))*((b*e-a*f)*(c+d*x)/((b*c-a*d)*(e+f*x)))^(-n)*
    Hypergeometric2F1[m+1,-n,m+2,-(d*e-c*f)*(a+b*x)/((b*c-a*d)*(e+f*x))] /;
FreeQ[{a,b,c,d,e,f,m,n,p},x] && EqQ[m+n+p+2,0] && Not[IntegerQ[n]]

5:  ∫(((((a+b x))^m) (((c+d x))^n))/(e+f x))dx  when  m+n+1∈(SuperPlus[ℤ])

Derivation: Algebraic expansion

Basis: (((((a+b x))^m) (((c+d x))^n))/(e+f x))==(((((c f-d e))^(m+n+1)) (((a+b x))^m))/((f^(m+n+1)) (((c+d x))^(m+1)) (e+f x)))+((((a+b x))^m)/((f^(m+n+1)) (((c+d x))^(m+1))))(((f^(m+n+1)) (((c+d x))^(m+n+1))-(((c f-d e))^(m+n+1)))/(e+f x))

Note: If  m+n+1∈(SuperPlus[ℤ]), then (((f^(m+n+1)) (((c+d x))^(m+n+1))-(((c f-d e))^(m+n+1)))/(e+f x)) is a polynomial in x.

Rule 1.1.1.3.24.5: If  m+n+1∈(SuperPlus[ℤ]), then

∫(((((a+b x))^m) (((c+d x))^n))/(e+f x))dx  ⟶  ((((c f-d e))^(m+n+1))/(f^(m+n+1))) ∫((((a+b x))^m)/((((c+d x))^(m+1)) (e+f x)))dx+(1/(f^(m+n+1))) ∫((((a+b x))^m)/(((c+d x))^(m+1)))(((f^(m+n+1)) (((c+d x))^(m+n+1))-(((c f-d e))^(m+n+1)))/(e+f x))dx

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_/(e_.+f_.*x_),x_Symbol] :=
  (c*f-d*e)^(m+n+1)/f^(m+n+1)*Int[(a+b*x)^m/((c+d*x)^(m+1)*(e+f*x)),x] + 
  1/f^(m+n+1)*Int[(a+b*x)^m/(c+d*x)^(m+1)*ExpandToSum[(f^(m+n+1)*(c+d*x)^(m+n+1)-(c*f-d*e)^(m+n+1))/(e+f*x),x],x] /;
FreeQ[{a,b,c,d,e,f,m,n},x] && IGtQ[m+n+1,0] && (LtQ[m,0] || SumSimplerQ[m,1] || Not[LtQ[n,0] || SumSimplerQ[n,1]])

6:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m+n+p+2∈(SuperMinus[ℤ])∧ m≠-1

Derivation: Nondegenerate trilinear recurrence 3 with A=1 and B=0

Note: If  m+n+p+2∈(SuperMinus[ℤ]), then ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx can be expressed in terms of the hypergeometric function 2F1.

Rule 1.1.1.3.24.6: If  m+n+p+2∈(SuperMinus[ℤ])∧ m≠-1, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
((b(((a+b x))^(m+1)) (((c+d x))^(n+1))(((e+f x))^(p+1)))/((m+1) (b c-a d) (b e-a f)))+
(1/((m+1) (b c-a d) (b e-a f)))∫(((a+b x))^(m+1)) (((c+d x))^n) (((e+f x))^p)(a d f (m+1)-b (d e (m+n+2)+c f (m+p+2))-b d f(m+n+p+3)x)dx

Program code:

Int[(a_.+b_.*x_)^m_*(c_.+d_.*x_)^n_.*(e_.+f_.*x_)^p_.,x_Symbol] :=
  b*(a+b*x)^(m+1)*(c+d*x)^(n+1)*(e+f*x)^(p+1)/((m+1)*(b*c-a*d)*(b*e-a*f)) + 
  1/((m+1)*(b*c-a*d)*(b*e-a*f))*Int[(a+b*x)^(m+1)*(c+d*x)^n*(e+f*x)^p*
    Simp[a*d*f*(m+1)-b*(d*e*(m+n+2)+c*f*(m+p+2))-b*d*f*(m+n+p+3)*x,x],x] /;
FreeQ[{a,b,c,d,e,f,m,n,p},x] && ILtQ[m+n+p+2,0] && NeQ[m,-1] && (SumSimplerQ[m,1] || Not[SumSimplerQ[n,1]] && Not[SumSimplerQ[p,1]])

25:  ∫(((a+b x))^m) (((c+d x))^n)(((f x))^p)dx  when  b c+a d==0 ∧ m-n∈(SuperPlus[ℤ])

Derivation: Algebraic expansion

Note: Integrals of this form can be expressed in terms of the confluent hypergeometric function 2F 1 instead of requiring the Appell hypergeometric function.

Rule 1.1.1.3.25: If  b c+a d==0 ∧ m-n∈(SuperPlus[ℤ]), then

∫(((a+b x))^m) (((c+d x))^n)(((f x))^p)dx  ⟶  ∫(((a+b x))^n)(((c+d x))^n)(((f x))^p)ExpandIntegrand[(((a+b x))^(m-n)) ,x]dx

Program code:

Int[(a_.+b_.*x_)^m_.*(c_.+d_.*x_)^n_.*(f_.*x_)^p_.,x_Symbol] :=
   Int[ExpandIntegrand[(a+b*x)^n*(c+d*x)^n*(f*x)^p,(a+b*x)^(m-n),x],x] /;
FreeQ[{a,b,c,d,f,m,n,p},x] && EqQ[b*c+a*d,0] && IGtQ[m-n,0] && NeQ[m+n+p+2,0]

A.  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ

1.  ∫(((b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ

1:  ∫(((b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ ∧ c>0 ∧ (p∈ℤ ∨ e>0)

Rule 1.1.1.3.A.1.1: If  m∉ℤ ∧ n∉ℤ ∧ c>0 ∧ (p∈ℤ ∨ e>0), then

∫(((b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  (((c^n) (e^p) (((b x))^(m+1)))/(b(m+1))) AppellF1[m+1,-n,-p,m+2,-((d x)/c),-((f x)/e)]

Program code:

Int[(b_.*x_)^m_*(c_+d_.*x_)^n_*(e_+f_.*x_)^p_,x_Symbol] :=
  c^n*e^p*(b*x)^(m+1)/(b*(m+1))*AppellF1[m+1,-n,-p,m+2,-d*x/c,-f*x/e] /;
FreeQ[{b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && GtQ[c,0] && (IntegerQ[p] || GtQ[e,0])

2:  ∫(((b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ ∧ -(d/(b c))>0 ∧ (p∈ℤ ∨ (d/(d e-c f))>0)

Rule 1.1.1.3.A.1.2: If  m∉ℤ ∧ n∉ℤ ∧ -(d/(b c))>0 ∧ (p∈ℤ ∨ (d/(d e-c f))>0), then

∫(((b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
((((c+d x))^(n+1))/(d (n+1) (((-(d/(b c))))^m) ((((d/(d e-c f))))^p))) AppellF1[n+1,-m,-p,n+2,1+((d x)/c),-((f (c+d x))/(d e-c f))]

Program code:

Int[(b_.*x_)^m_*(c_+d_.*x_)^n_*(e_+f_.*x_)^p_,x_Symbol] :=
  (c+d*x)^(n+1)/(d*(n+1)*(-d/(b*c))^m*(d/(d*e-c*f))^p)*AppellF1[n+1,-m,-p,n+2,1+d*x/c,-f*(c+d*x)/(d*e-c*f)] /;
FreeQ[{b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && GtQ[-d/(b*c),0] && (IntegerQ[p] || GtQ[d/(d*e-c*f),0])

3:  ∫(((b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ ∧ c≯0

Derivation: Piecewise constant extraction

Basis: (∂_x)((((c+d x))^n)/(((((c+d x)/c)))^n))==0

Rule 1.1.1.3.A.1.3: If  m∉ℤ ∧ n∉ℤ ∧ c≯0, then

∫(((b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  (((c^(IntPart[n])) (((c+d x))^(FracPart[n])))/(((1+((d x)/c)))^(FracPart[n]))) ∫(((b x))^m) (((1+((d x)/c)))^n) (((e+f x))^p)dx

Program code:

Int[(b_.*x_)^m_*(c_+d_.*x_)^n_*(e_+f_.*x_)^p_,x_Symbol] :=
  c^IntPart[n]*(c+d*x)^FracPart[n]/(1+d*x/c)^FracPart[n]*Int[(b*x)^m*(1+d*x/c)^n*(e+f*x)^p,x] /;
FreeQ[{b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[GtQ[c,0]]

2.  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ ∧ p∈ℤ

1:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ ∧ p∈ℤ ∧ (b/(b c-a d))>0

Rule 1.1.1.3.A.2.1: If  m∉ℤ ∧ n∉ℤ ∧ p∈ℤ ∧ (b/(b c-a d))>0, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
(((((b e-a f))^p) (((a+b x))^(m+1)))/((b^(p+1))(m+1) ((((b/(b c-a d))))^n))) AppellF1[m+1,-n,-p,m+2,-((d (a+b x))/(b c-a d)),-((f (a+b x))/(b e-a f))]

Program code:

Int[(a_+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] :=
  (b*e-a*f)^p*(a+b*x)^(m+1)/(b^(p+1)*(m+1)*(b/(b*c-a*d))^n)*
    AppellF1[m+1,-n,-p,m+2,-d*(a+b*x)/(b*c-a*d),-f*(a+b*x)/(b*e-a*f)] /;
FreeQ[{a,b,c,d,e,f,m,n},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && IntegerQ[p] && GtQ[b/(b*c-a*d),0] && 
  Not[GtQ[d/(d*a-c*b),0] && SimplerQ[c+d*x,a+b*x]]

2:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ ∧ p∈ℤ ∧ (b/(b c-a d))≯0

Derivation: Piecewise constant extraction

Basis: (∂_x)((((c+d x))^n)/(((((b (c+d x))/(b c-a d))))^n))==0

Rule 1.1.1.3.A.2.2: If  m∉ℤ ∧ n∉ℤ ∧ p∈ℤ ∧ (b/(b c-a d))≯0, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  ((((c+d x))^(FracPart[n]))/(((((b/(b c-a d))))^(IntPart[n]))(((((b (c+d x))/(b c-a d))))^(FracPart[n])))) ∫(((a+b x))^m) (((((b c)/(b c-a d))+((b d x)/(b c-a d))))^n) (((e+f x))^p)dx

Program code:

Int[(a_+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] :=
  (c+d*x)^FracPart[n]/((b/(b*c-a*d))^IntPart[n]*(b*(c+d*x)/(b*c-a*d))^FracPart[n])*
    Int[(a+b*x)^m*(b*c/(b*c-a*d)+b*d*x/(b*c-a*d))^n*(e+f*x)^p,x] /;
FreeQ[{a,b,c,d,e,f,m,n},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && IntegerQ[p] && Not[GtQ[b/(b*c-a*d),0]] && 
  Not[SimplerQ[c+d*x,a+b*x]]

3.  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ ∧ p∉ℤ

1.  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ ∧ p∉ℤ ∧ (b/(b c-a d))>0

1:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ ∧ p∉ℤ ∧ (b/(b c-a d))>0 ∧ (b/(b e-a f))>0

Rule 1.1.1.3.A.3.1.1: If  m∉ℤ ∧ n∉ℤ ∧ p∉ℤ ∧ (b/(b c-a d))>0 ∧ (b/(b e-a f))>0, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  
((((a+b x))^(m+1))/(b(m+1) ((((b/(b c-a d))))^n) ((((b/(b e-a f))))^p))) AppellF1[m+1,-n,-p,m+2,-((d (a+b x))/(b c-a d)),-((f (a+b x))/(b e-a f))]

Program code:

Int[(a_+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] :=
  (a+b*x)^(m+1)/(b*(m+1)*(b/(b*c-a*d))^n*(b/(b*e-a*f))^p)*AppellF1[m+1,-n,-p,m+2,-d*(a+b*x)/(b*c-a*d),-f*(a+b*x)/(b*e-a*f)] /;
FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && 
  GtQ[b/(b*c-a*d),0] && GtQ[b/(b*e-a*f),0] && 
  Not[GtQ[d/(d*a-c*b),0] && GtQ[d/(d*e-c*f),0] && SimplerQ[c+d*x,a+b*x]] && 
  Not[GtQ[f/(f*a-e*b),0] && GtQ[f/(f*c-e*d),0] && SimplerQ[e+f*x,a+b*x]]

2:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ ∧ p∉ℤ ∧ (b/(b c-a d))>0 ∧ (b/(b e-a f))≯0

Derivation: Piecewise constant extraction

Basis: (∂_x)((((e+f x))^p)/(((((b (e+f x))/(b e-a f))))^p))==0

Rule 1.1.1.3.A.3.1.2: If  m∉ℤ ∧ n∉ℤ ∧ p∉ℤ ∧ (b/(b c-a d))>0 ∧ (b/(b e-a f))≯0, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  ((((e+f x))^(FracPart[p]))/(((((b/(b e-a f))))^(IntPart[p]))(((((b (e+f x))/(b e-a f))))^(FracPart[p])))) ∫(((a+b x))^m) (((c+d x))^n) (((((b e)/(b e-a f))+((b f x)/(b e-a f))))^p)dx

Program code:

Int[(a_+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] :=
  (e+f*x)^FracPart[p]/((b/(b*e-a*f))^IntPart[p]*(b*(e+f*x)/(b*e-a*f))^FracPart[p])*
    Int[(a+b*x)^m*(c+d*x)^n*(b*e/(b*e-a*f)+b*f*x/(b*e-a*f))^p,x] /;
FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && 
  GtQ[b/(b*c-a*d),0] && Not[GtQ[b/(b*e-a*f),0]]

2:  ∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  when  m∉ℤ ∧ n∉ℤ ∧ p∉ℤ ∧ (b/(b c-a d))≯0

Derivation: Piecewise constant extraction

Basis: (∂_x)((((c+d x))^n)/(((((b (c+d x))/(b c-a d))))^n))==0

Rule 1.1.1.3.A.3.2: If  m∉ℤ ∧ n∉ℤ ∧ p∉ℤ ∧ (b/(b c-a d))≯0, then

∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx  ⟶  ((((c+d x))^(FracPart[n]))/(((((b/(b c-a d))))^(IntPart[n]))(((((b (c+d x))/(b c-a d))))^(FracPart[n])))) ∫(((a+b x))^m) (((((b c)/(b c-a d))+((b d x)/(b c-a d))))^n) (((e+f x))^p)dx

Program code:

Int[(a_+b_.*x_)^m_*(c_.+d_.*x_)^n_*(e_.+f_.*x_)^p_,x_Symbol] :=
  (c+d*x)^FracPart[n]/((b/(b*c-a*d))^IntPart[n]*(b*(c+d*x)/(b*c-a*d))^FracPart[n])*
    Int[(a+b*x)^m*(b*c/(b*c-a*d)+b*d*x/(b*c-a*d))^n*(e+f*x)^p,x] /;
FreeQ[{a,b,c,d,e,f,m,n,p},x] && Not[IntegerQ[m]] && Not[IntegerQ[n]] && Not[IntegerQ[p]] && Not[GtQ[b/(b*c-a*d),0]] && 
  Not[SimplerQ[c+d*x,a+b*x]] && Not[SimplerQ[e+f*x,a+b*x]]

S:  ∫(((a+b u))^m) (((c+d u))^n)(((e+f u))^p)dx  when  u==g+h x

Derivation: Integration by substitution

Rule 1.1.1.3.S: If  u==g+h x, then

∫(((a+b u))^m) (((c+d u))^n)(((e+f u))^p)dx  ⟶  (1/h)Subst[∫(((a+b x))^m) (((c+d x))^n) (((e+f x))^p)dx,x,u]

Program code:

Int[(a_.+b_.*u_)^m_.*(c_.+d_.*u_)^n_.*(e_+f_.*u_)^p_.,x_Symbol] :=
  1/Coefficient[u,x,1]*Subst[Int[(a+b*x)^m*(c+d*x)^n*(e+f*x)^p,x],x,u] /;
FreeQ[{a,b,c,d,e,f,m,n,p},x] && LinearQ[u,x] && NeQ[u,x]