Rules for integrands of the formP[x] (((f x))^m) (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)

1.  ∫(((x^m) (A+B (x^2)+C (x^4)))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ (m/2)∈ℤ

1:  ∫(((x^m) (A+B (x^2)+C (x^4)))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ (m/2)∈(SuperPlus[ℤ])

Rule: If  (b^2)-4 a c≠0 ∧ (m/2)∈(SuperPlus[ℤ]), then

∫(((x^m) (A+B (x^2)+C (x^4)))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  ⟶  

((C (x^(m-1)) (Sqrt[a+b (x^2)+c (x^4)]))/(c e (m+1)))-(1/(c e (m+1))) ∫((x^(m-2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))·
(a C d (m-1)-(A c e (m+1)-C (a e (m-1)+b d m)) (x^2)-(B c e (m+1)-C (b e m+c d (m+1))) (x^4))dx

Program code:

Int[Px_*x_^m_/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  With[{A=Coeff[Px,x,0],B=Coeff[Px,x,2],C=Coeff[Px,x,4]},
  C*x^(m-1)*Sqrt[a+b*x^2+c*x^4]/(c*e*(m+1)) - 
  1/(c*e*(m+1))*Int[(x^(m-2)/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]))*
    Simp[a*C*d*(m-1)-(A*c*e*(m+1)-C*(a*e*(m-1)+b*d*m))*x^2-(B*c*e*(m+1)-C*(b*e*m+c*d*(m+1)))*x^4,x],x]] /;
FreeQ[{a,b,c,d,e},x] && PolyQ[Px,x^2,2] && NeQ[b^2-4*a*c,0] && IGtQ[m/2,0]

Int[Px_*x_^m_/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  With[{A=Coeff[Px,x,0],B=Coeff[Px,x,2],C=Coeff[Px,x,4]},
  C*x^(m-1)*Sqrt[a+c*x^4]/(c*e*(m+1)) - 
  1/(c*e*(m+1))*Int[(x^(m-2)/((d+e*x^2)*Sqrt[a+c*x^4]))*
    Simp[a*C*d*(m-1)-(A*c*e*(m+1)-C*a*e*(m-1))*x^2-(B*c*e*(m+1)-C*c*d*(m+1))*x^4,x],x]] /;
FreeQ[{a,c,d,e},x] && PolyQ[Px,x^2,2] && IGtQ[m/2,0]

2:  ∫(((x^m) (A+B (x^2)+C (x^4)))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ (m/2)∈(SuperMinus[ℤ])

Rule: If  (b^2)-4 a c≠0 ∧ (m/2)∈(SuperMinus[ℤ]), then

∫(((x^m) (A+B (x^2)+C (x^4)))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  ⟶  

((A (x^(m+1)) (Sqrt[a+b (x^2)+c (x^4)]))/(a d (m+1)))+(1/(a d (m+1))) ∫((x^(m+2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))·
(a B d (m+1)-A (a e (m+1)+b d (m+2))+(a C d (m+1)-A (b e (m+2)+c d (m+3))) (x^2)-A c e (m+3) (x^4))dx

Program code:

Int[Px_*x_^m_/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  With[{A=Coeff[Px,x,0],B=Coeff[Px,x,2],C=Coeff[Px,x,4]},
  A*x^(m+1)*Sqrt[a+b*x^2+c*x^4]/(a*d*(m+1)) + 
  1/(a*d*(m+1))*Int[(x^(m+2)/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]))*
    Simp[a*B*d*(m+1)-A*(a*e*(m+1)+b*d*(m+2))+(a*C*d*(m+1)-A*(b*e*(m+2)+c*d*(m+3)))*x^2-A*c*e*(m+3)*x^4,x],x]] /;
FreeQ[{a,b,c,d,e},x] && PolyQ[Px,x^2,2] && NeQ[b^2-4*a*c,0] && ILtQ[m/2,0]

Int[Px_*x_^m_/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  With[{A=Coeff[Px,x,0],B=Coeff[Px,x,2],C=Coeff[Px,x,4]},
  A*x^(m+1)*Sqrt[a+c*x^4]/(a*d*(m+1)) + 
  1/(a*d*(m+1))*Int[(x^(m+2)/((d+e*x^2)*Sqrt[a+c*x^4]))*
    Simp[a*B*d*(m+1)-A*a*e*(m+1)+(a*C*d*(m+1)-A*c*d*(m+3))*x^2-A*c*e*(m+3)*x^4,x],x]] /;
FreeQ[{a,c,d,e},x] && PolyQ[Px,x^2,2] && ILtQ[m/2,0]

Rules for integrands of the formP[x] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)

1:  ∫x P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx

Derivation: Integration by substitution

Basis: x F[(x^2)]==(1/2) Subst[F[x],x,(x^2)] (∂_x)(x^2)

Rule 1.2.2.7.1:

∫x P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  ⟶  (1/2) Subst[∫P[x](((d+e x))^q) (((a+b x+c (x^2)))^p)dx,x,(x^2)]

Program code:

Int[x_*Px_*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] :=
  1/2*Subst[Int[ReplaceAll[Px,x->Sqrt[x]]*(d+e*x)^q*(a+b*x+c*x^2)^p,x],x,x^2] /;
FreeQ[{a,b,c,d,e,p,q},x] && PolyQ[Px,x^2]

2:  ∫(Subscript[P, r])[x] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  when PolynomialRemainder[(Subscript[P, r])[x],x,x]==0

Derivation: Algebraic simplification

Rule 1.2.2.7.2: If  PolynomialRemainder[(Subscript[P, r])[x],x,x]==0, then

∫(Subscript[P, r])[x] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  ⟶  ∫x PolynomialQuotient[(Subscript[P, r])[x],x,x] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx

Program code:

Int[Pr_*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] :=
  Int[x*PolynomialQuotient[Pr,x,x]*(d+e*x^2)^q*(a+b*x^2+c*x^4)^p,x] /;
FreeQ[{a,b,c,d,e,p,q},x] && PolyQ[Pr,x] && EqQ[PolynomialRemainder[Pr,x,x],0] && Not[MatchQ[Pr,x^m_.*u_. /; IntegerQ[m]]]

3:  ∫(Subscript[P, r])[x] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  when ¬(Subscript[P, r])[(x^2)]

Derivation: Algebraic expansion

Basis: (Subscript[P, r])[x]==(∑_(k=0)^(r/2))(Subscript[P, r])[x,2 k] (x^(2 k))+x (∑_(k=0)^((r-1)/2))(Subscript[P, r])[x,2 k+1] (x^(2 k))

Note: This rule transforms (Subscript[P, r])[x] into a sum of the form (Subscript[Q, s])[(x^2)]+x (Subscript[R, t])[(x^2)].

Rule 1.2.2.7.3: If ¬(Subscript[P, r])[(x^2)], then

∫(Subscript[P, r])[x] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  ⟶  
∫ ((∑_(k=0)^(r/2))(Subscript[P, r])[x,2 k] (x^(2 k))) (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx+∫ x ((∑_(k=0)^((r-1)/2))(Subscript[P, r])[x,2 k+1] (x^(2 k))) (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx

Program code:

Int[Pr_*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] :=
  Module[{r=Expon[Pr,x],k},
  Int[Sum[Coeff[Pr,x,2*k]*x^(2*k),{k,0,r/2}]*(d+e*x^2)^q*(a+b*x^2+c*x^4)^p,x] + 
  Int[x*Sum[Coeff[Pr,x,2*k+1]*x^(2*k),{k,0,(r-1)/2}]*(d+e*x^2)^q*(a+b*x^2+c*x^4)^p,x]] /;
FreeQ[{a,b,c,d,e,p,q},x] && PolyQ[Pr,x] && Not[PolyQ[Pr,x^2]]

4.  ∫P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)==0

1:  ∫P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)==0 ∧ p∈ℤ

Derivation: Algebraic simplification

Basis: If  c (d^2)-b d e+a (e^2)==0, then a+b z+c (z^2)==(d+e z) ((a/d)+((c z)/e))

Rule 1.2.2.7.4.1: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)==0 ∧ p∈ℤ, then

∫P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  ⟶  ∫P[(x^2)] (((d+e (x^2)))^(p+q)) ((((a/d)+((c (x^2))/e)))^p)dx

Program code:

Int[Px_*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] :=
  Int[Px*(d+e*x^2)^(p+q)*(a/d+c/e*x^2)^p,x] /;
FreeQ[{a,b,c,d,e,q},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && IntegerQ[p] && 
  (PolyQ[Px,x^2] || MatchQ[Px,(f_+g_.*x^2)^r_./;FreeQ[{f,g,r},x]])

Int[Px_*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] :=
  Int[Px*(d+e*x^2)^(p+q)*(a/d+c/e*x^2)^p,x] /;
FreeQ[{a,c,d,e,q},x] && EqQ[c*d^2+a*e^2,0] && IntegerQ[p] && 
  (PolyQ[Px,x^2] || MatchQ[Px,(f_+g_.*x^2)^r_./;FreeQ[{f,g,r},x]])

2:  ∫P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)==0 ∧ p∉ℤ

Derivation: Piecewise constant extraction

Basis: If  c (d^2)-b d e+a (e^2)==0, then (∂_x)((((a+b (x^2)+c (x^4)))^p)/((((d+e (x^2)))^p) ((((a/d)+((c (x^2))/e)))^p)))==0

Basis: If  c (d^2)-b d e+a (e^2)==0, then ((((a+b (x^2)+c (x^4)))^p)/((((d+e (x^2)))^p) ((((a/d)+((c (x^2))/e)))^p)))==((((a+b (x^2)+c (x^4)))^(FracPart[p]))/((((d+e (x^2)))^(FracPart[p])) ((((a/d)+((c (x^2))/e)))^(FracPart[p]))))

Rule 1.2.2.7.4.2: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)==0 ∧ p∉ℤ, then

∫P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  ⟶  ((((a+b (x^2)+c (x^4)))^(FracPart[p]))/((((d+e (x^2)))^(FracPart[p])) ((((a/d)+((c (x^2))/e)))^(FracPart[p]))))∫P[(x^2)] (((d+e (x^2)))^(p+q)) ((((a/d)+((c (x^2))/e)))^p)dx

Program code:

Int[Px_*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] :=
  (a+b*x^2+c*x^4)^FracPart[p]/((d+e*x^2)^FracPart[p]*(a/d+c*x^2/e)^FracPart[p])*
    Int[Px*(d+e*x^2)^(p+q)*(a/d+c/e*x^2)^p,x] /;
FreeQ[{a,b,c,d,e,p,q},x] && NeQ[b^2-4*a*c,0] && EqQ[c*d^2-b*d*e+a*e^2,0] && Not[IntegerQ[p]] && 
  (PolyQ[Px,x^2] || MatchQ[Px,(f_+g_.*x^2)^r_./;FreeQ[{f,g,r},x]])

Int[Px_*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_,x_Symbol] :=
  (a+c*x^4)^FracPart[p]/((d+e*x^2)^FracPart[p]*(a/d+c*x^2/e)^FracPart[p])*
    Int[Px*(d+e*x^2)^(p+q)*(a/d+c/e*x^2)^p,x] /;
FreeQ[{a,c,d,e,p,q},x] && EqQ[c*d^2+a*e^2,0] && Not[IntegerQ[p]] && 
  (PolyQ[Px,x^2] || MatchQ[Px,(f_+g_.*x^2)^r_./;FreeQ[{f,g,r},x]])

5:  ∫P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ p∈ℤ

Derivation: Algebraic expansion

Rule 1.2.2.7.5: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ q∈ℤ ∧ p∈ℤ, then

∫P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  ⟶  ∫ExpandIntegrand[P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p),x] dx

Program code:

Int[Px_*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] :=
  Int[ExpandIntegrand[Px*(d+e*x^2)^q*(a+b*x^2+c*x^4)^p,x],x] /;
FreeQ[{a,b,c,d,e,q},x] && PolyQ[Px,x^2] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && IntegerQ[p]

Int[Px_*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] :=
  Int[ExpandIntegrand[Px*(d+e*x^2)^q*(a+c*x^4)^p,x],x] /;
FreeQ[{a,c,d,e,q},x] && PolyQ[Px,x^2] && NeQ[c*d^2+a*e^2,0] && IntegerQ[p]

6.  ∫P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ p+(1/2)∈ℤ ∧ q∈ℤ

1.  ∫((P[(x^2)] (((d+e (x^2)))^q))/(Sqrt[a+b (x^2)+c (x^4)]))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ q∈ℤ

1:  ∫(((((d+e (x^2)))^q) (A+B (x^2)+C (x^4)))/(Sqrt[a+b (x^2)+c (x^4)]))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ q∈(SuperPlus[ℤ])

Rule 1.2.2.7.6.1.1: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ q∈(SuperPlus[ℤ]), then

∫(((((d+e (x^2)))^q) (A+B (x^2)+C (x^4)))/(Sqrt[a+b (x^2)+c (x^4)]))dx  ⟶  

((C x (((d+e (x^2)))^q) (Sqrt[a+b (x^2)+c (x^4)]))/(c (2 q+3)))+
(1/(c (2 q+3))) ∫(((((d+e (x^2)))^(q-1)) (A c d (2 q+3)-a C d+(c (B d+A e) (2 q+3)-C (2 b d+a e+2 a e q)) (x^2)+(B c e (2 q+3)-2 C (b e-c d q+b e q)) (x^4)))/(Sqrt[a+b (x^2)+c (x^4)]))dx

Program code:

Int[(d_+e_.*x_^2)^q_*P4x_/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] :=
  With[{A=Coeff[P4x,x,0],B=Coeff[P4x,x,2],C=Coeff[P4x,x,4]},
  C*x*(d+e*x^2)^q*Sqrt[a+b*x^2+c*x^4]/(c*(2*q+3)) + 
  1/(c*(2*q+3))*Int[(d+e*x^2)^(q-1)/Sqrt[a+b*x^2+c*x^4]*
    Simp[A*c*d*(2*q+3)-a*C*d+(c*(B*d+A*e)*(2*q+3)-C*(2*b*d+a*e+2*a*e*q))*x^2+(B*c*e*(2*q+3)-2*C*(b*e-c*d*q+b*e*q))*x^4,x],x]] /;
FreeQ[{a,b,c,d,e},x] && PolyQ[P4x,x^2] && EqQ[Expon[P4x,x],4] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && IGtQ[q,0]

Int[(d_+e_.*x_^2)^q_*P4x_/Sqrt[a_+c_.*x_^4],x_Symbol] :=
  With[{A=Coeff[P4x,x,0],B=Coeff[P4x,x,2],C=Coeff[P4x,x,4]},
  C*x*(d+e*x^2)^q*Sqrt[a+c*x^4]/(c*(2*q+3)) + 
  1/(c*(2*q+3))*Int[(d+e*x^2)^(q-1)/Sqrt[a+c*x^4]*
    Simp[A*c*d*(2*q+3)-a*C*d+(c*(B*d+A*e)*(2*q+3)-a*C*e*(2*q+1))*x^2+(B*c*e*(2*q+3)+2*c*C*d*q)*x^4,x],x]] /;
FreeQ[{a,c,d,e},x] && PolyQ[P4x,x^2] && EqQ[Expon[P4x,x],4] && NeQ[c*d^2+a*e^2,0] && IGtQ[q,0]

2:  ∫(((((d+e (x^2)))^q) (A+B (x^2)+C (x^4)))/(Sqrt[a+b (x^2)+c (x^4)]))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ q+1∈(SuperMinus[ℤ])

Rule 1.2.2.7.6.1.2: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ q+1∈(SuperMinus[ℤ]), then

∫(((((d+e (x^2)))^q) (A+B (x^2)+C (x^4)))/(Sqrt[a+b (x^2)+c (x^4)]))dx  ⟶  

-(((C (d^2)-B d e+A (e^2)) x (((d+e (x^2)))^(q+1)) (Sqrt[a+b (x^2)+c (x^4)]))/(2 d (q+1) (c (d^2)-b d e+a (e^2))))+(1/(2 d (q+1) (c (d^2)-b d e+a (e^2))))∫((((d+e (x^2)))^(q+1))/(Sqrt[a+b (x^2)+c (x^4)]))·
(a d (C d-B e)+A (a (e^2) (2 q+3)+2 d (c d-b e) (q+1))-2 ((B d-A e) (b e (q+2)-c d (q+1))-C d (b d+a e (q+1))) (x^2)+c (C (d^2)-B d e+A (e^2)) (2 q+5) (x^4))dx

Program code:

Int[(d_+e_.*x_^2)^q_*P4x_/Sqrt[a_+b_.*x_^2+c_.*x_^4],x_Symbol] :=
  With[{A=Coeff[P4x,x,0],B=Coeff[P4x,x,2],C=Coeff[P4x,x,4]},
  -(C*d^2-B*d*e+A*e^2)*x*(d+e*x^2)^(q+1)*Sqrt[a+b*x^2+c*x^4]/(2*d*(q+1)*(c*d^2-b*d*e+a*e^2)) + 
  1/(2*d*(q+1)*(c*d^2-b*d*e+a*e^2))*Int[(d+e*x^2)^(q+1)/Sqrt[a+b*x^2+c*x^4]*
    Simp[a*d*(C*d-B*e)+A*(a*e^2*(2*q+3)+2*d*(c*d-b*e)*(q+1))-
      2*((B*d-A*e)*(b*e*(q+2)-c*d*(q+1))-C*d*(b*d+a*e*(q+1)))*x^2+
      c*(C*d^2-B*d*e+A*e^2)*(2*q+5)*x^4,x],x]] /;
FreeQ[{a,b,c,d,e},x] && PolyQ[P4x,x^2] && LeQ[Expon[P4x,x],4] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && ILtQ[q,-1]

Int[(d_+e_.*x_^2)^q_*P4x_/Sqrt[a_+c_.*x_^4],x_Symbol] :=
  With[{A=Coeff[P4x,x,0],B=Coeff[P4x,x,2],C=Coeff[P4x,x,4]},
  -(C*d^2-B*d*e+A*e^2)*x*(d+e*x^2)^(q+1)*Sqrt[a+c*x^4]/(2*d*(q+1)*(c*d^2+a*e^2)) + 
  1/(2*d*(q+1)*(c*d^2+a*e^2))*Int[(d+e*x^2)^(q+1)/Sqrt[a+c*x^4]*
    Simp[a*d*(C*d-B*e)+A*(a*e^2*(2*q+3)+2*c*d^2*(q+1))+2*d*(B*c*d-A*c*e+a*C*e)*(q+1)*x^2+c*(C*d^2-B*d*e+A*e^2)*(2*q+5)*x^4,x],x]] /;
FreeQ[{a,c,d,e},x] && PolyQ[P4x,x^2] && LeQ[Expon[P4x,x],4] && NeQ[c*d^2+a*e^2,0] && ILtQ[q,-1]

3.  ∫((P[(x^2)])/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0

1.  ∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0

1.  ∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)==0

1:  ∫((A+B (x^2))/((d+e (x^2))(Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)==0 ∧ B d+A e==0

Derivation: Integration by substitution

Basis: If  c (d^2)-a (e^2)==0 ∧ B d+A e==0, then ((A+B (x^2))/((d+e (x^2))(Sqrt[a+b (x^2)+c (x^4)])))==A Subst[(1/(d-(b d-2 a e) (x^2))),x,(x/(Sqrt[a+b (x^2)+c (x^4)]))] (∂_x)(x/(Sqrt[a+b (x^2)+c (x^4)]))

Rule 1.2.2.7.6.1.3.1.1.1: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)==0 ∧ B d+A e==0, then

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

Program code:

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

Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  A*Subst[Int[1/(d+2*a*e*x^2),x],x,x/Sqrt[a+c*x^4]] /;
FreeQ[{a,c,d,e,A,B},x] && NeQ[c*d^2+a*e^2,0] && EqQ[c*d^2-a*e^2,0] && EqQ[B*d+A*e,0]

2:  ∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)==0 ∧ B d+A e≠0

Derivation: Algebraic expansion

Rule 1.2.2.7.6.1.3.1.1.2: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)==0 ∧ B d+A e≠0, then

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

Program code:

Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  (B*d+A*e)/(2*d*e)*Int[1/Sqrt[a+b*x^2+c*x^4],x] - 
  (B*d-A*e)/(2*d*e)*Int[(d-e*x^2)/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x] /;
FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && EqQ[c*d^2-a*e^2,0] && NeQ[B*d+A*e,0]

Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  (B*d+A*e)/(2*d*e)*Int[1/Sqrt[a+c*x^4],x] - 
  (B*d-A*e)/(2*d*e)*Int[(d-e*x^2)/((d+e*x^2)*Sqrt[a+c*x^4]),x] /;
FreeQ[{a,c,d,e,A,B},x] && NeQ[c*d^2+a*e^2,0] && EqQ[c*d^2-a*e^2,0] && NeQ[B*d+A*e,0]

2.  ∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (Sqrt[(b^2)-4 a c])∈ℝ ∨ c (A^2)-b A B+a (B^2)==0

1:  ∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (A^2)-b A B+a (B^2)==0

Derivation: Piecewise constant extraction

Basis: If  c (A^2)-b A B+a (B^2)==0, then (∂_x)(((Sqrt[A+B (x^2)]) (Sqrt[(a/A)+((c (x^2))/B)]))/(Sqrt[a+b (x^2)+c (x^4)]))==0

Rule 1.2.2.7.6.1.3.1.2.1: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (A^2)-b A B+a (B^2)==0, then

∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  ⟶  (((Sqrt[A+B (x^2)]) (Sqrt[(a/A)+((c (x^2))/B)]))/(Sqrt[a+b (x^2)+c (x^4)])) ∫((Sqrt[A+B (x^2)])/((d+e (x^2)) (Sqrt[(a/A)+((c (x^2))/B)])))dx

Program code:

Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  Sqrt[A+B*x^2]*Sqrt[a/A+c*x^2/B]/Sqrt[a+b*x^2+c*x^4]*Int[Sqrt[A+B*x^2]/((d+e*x^2)*Sqrt[a/A+c*x^2/B]),x] /;
FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && EqQ[c*A^2-b*A*B+a*B^2,0]

Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  Sqrt[A+B*x^2]*Sqrt[a/A+c*x^2/B]/Sqrt[a+c*x^4]*Int[Sqrt[A+B*x^2]/((d+e*x^2)*Sqrt[a/A+c*x^2/B]),x] /;
FreeQ[{a,c,d,e,A,B},x] && NeQ[c*d^2+a*e^2,0] && EqQ[c*A^2+a*B^2,0]

2:  ∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c>0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (A^2)-b A B+a (B^2)≠0 ∧ (Sqrt[(b^2)-4 a c])∈ℝ

Derivation: Algebraic expansion

Note: If  q->(Sqrt[(b^2)-4 a c]) and c (d^2)-b d e+a (e^2)≠0, then 2 a e-d (b+q)≠0.

Rule 1.2.2.7.6.1.3.1.2.2: If  (b^2)-4 a c>0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (A^2)-b A B+a (B^2)≠0, let q->(Sqrt[(b^2)-4 a c]), if  q∈ℝ, then

∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  ⟶  ((2 a B-A (b+q))/(2 a e-d (b+q))) ∫(1/(Sqrt[a+b (x^2)+c (x^4)]))dx-((B d-A e)/(2 a e-d (b+q))) ∫((2 a+(b+q) (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx

Program code:

Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  With[{q=Sqrt[b^2-4*a*c]},
  (2*a*B-A*(b+q))/(2*a*e-d*(b+q))*Int[1/Sqrt[a+b*x^2+c*x^4],x] - 
  (B*d-A*e)/(2*a*e-d*(b+q))*Int[(2*a+(b+q)*x^2)/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x] /;
 RationalQ[q]] /;
FreeQ[{a,b,c,d,e,A,B},x] && GtQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && NeQ[c*A^2-b*A*B+a*B^2,0]

Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  With[{q=Sqrt[-a*c]},
  (a*B-A*q)/(a*e-d*q)*Int[1/Sqrt[a+c*x^4],x] - 
  (B*d-A*e)/(a*e-d*q)*Int[(a+q*x^2)/((d+e*x^2)*Sqrt[a+c*x^4]),x] /;
 RationalQ[q]] /;
FreeQ[{a,c,d,e,A,B},x] && GtQ[-a*c,0] && EqQ[c*d^2+a*e^2,0] && NeQ[c*A^2+a*B^2,0]

3.  ∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0

1.  ∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0 ∧ (c/a)>0

x:  ∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0 ∧ (c/a)>0 ∧ c (A^2)-a (B^2)==0

Rule 1.2.2.7.6.1.3.1.3.1.x: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0 ∧ (c/a)>0 ∧ c (A^2)-a (B^2)==0, let q->(Sqrt[(B/A)]), then

∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  ⟶  
-(((B d-A e) ArcTan[(((Sqrt[-b+((c d)/e)+((a e)/d)]) x)/(Sqrt[a+b (x^2)+c (x^4)]))])/(2 d e (Sqrt[-b+((c d)/e)+((a e)/d)])))+((B q (c (d^2)-a (e^2)) (A+B (x^2)) (Sqrt[(((A^2) (a+b (x^2)+c (x^4)))/(a (((A+B (x^2)))^2)))]))/(4 c d e (B d-A e) (Sqrt[a+b (x^2)+c (x^4)]))) EllipticPi[-((((B d-A e))^2)/(4 d e A B)),2 ArcTan[q x],(1/2)-((b A)/(4 a B))]

Program code:

(* Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  With[{q=Rt[B/A,2]},
  -(B*d-A*e)*ArcTan[Rt[-b+c*d/e+a*e/d,2]*x/Sqrt[a+b*x^2+c*x^4]]/(2*d*e*Rt[-b+c*d/e+a*e/d,2]) + 
  B*q*(c*d^2-a*e^2)*(A+B*x^2)*Sqrt[A^2*(a+b*x^2+c*x^4)/(a*(A+B*x^2)^2)]/(4*c*d*e*(B*d-A*e)*Sqrt[a+b*x^2+c*x^4])*
    EllipticPi[-(B*d-A*e)^2/(4*d*e*A*B),2*ArcTan[q*x],1/2-b*A/(4*a*B)]] /;
FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && NeQ[c*d^2-a*e^2,0] && PosQ[c/a] && EqQ[c*A^2-a*B^2,0] *)

(* Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  With[{q=Rt[B/A,2]},
  -(B*d-A*e)*ArcTan[Rt[c*d/e+a*e/d,2]*x/Sqrt[a+c*x^4]]/(2*d*e*Rt[c*d/e+a*e/d,2]) + 
  B*q*(c*d^2-a*e^2)*(A+B*x^2)*Sqrt[A^2*(a+c*x^4)/(a*(A+B*x^2)^2)]/(4*c*d*e*(B*d-A*e)*Sqrt[a+c*x^4])*
    EllipticPi[-(B*d-A*e)^2/(4*d*e*A*B),2*ArcTan[q*x],1/2]] /;
FreeQ[{a,c,d,e,A,B},x] && NeQ[c*d^2+a*e^2,0] && NeQ[c*d^2-a*e^2,0] && PosQ[c/a] && EqQ[c*A^2-a*B^2,0] *)

1:  ∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0 ∧ (c/a)>0 ∧ c (A^2)-a (B^2)==0

Rule 1.2.2.7.6.1.3.1.3.1.1: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0 ∧ (c/a)>0 ∧ c (A^2)-a (B^2)==0, let q->(Sqrt[(B/A)]), then

∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  ⟶  
-(((B d-A e) ArcTan[(((Sqrt[-b+((c d)/e)+((a e)/d)]) x)/(Sqrt[a+b (x^2)+c (x^4)]))])/(2 d e (Sqrt[-b+((c d)/e)+((a e)/d)])))+(((B d+A e) (A+B (x^2)) (Sqrt[(((A^2) (a+b (x^2)+c (x^4)))/(a (((A+B (x^2)))^2)))]))/(4 d e A q (Sqrt[a+b (x^2)+c (x^4)]))) EllipticPi[-((((B d-A e))^2)/(4 d e A B)),2 ArcTan[q x],(1/2)-((b A)/(4 a B))]

Program code:

Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  With[{q=Rt[B/A,2]},
  -(B*d-A*e)*ArcTan[Rt[-b+c*d/e+a*e/d,2]*x/Sqrt[a+b*x^2+c*x^4]]/(2*d*e*Rt[-b+c*d/e+a*e/d,2]) + 
  (B*d+A*e)*(A+B*x^2)*Sqrt[A^2*(a+b*x^2+c*x^4)/(a*(A+B*x^2)^2)]/(4*d*e*A*q*Sqrt[a+b*x^2+c*x^4])*
    EllipticPi[Cancel[-(B*d-A*e)^2/(4*d*e*A*B)],2*ArcTan[q*x],1/2-b*A/(4*a*B)]] /;
FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && NeQ[c*d^2-a*e^2,0] && PosQ[c/a] && EqQ[c*A^2-a*B^2,0]

Int[(A_+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  With[{q=Rt[B/A,2]},
  -(B*d-A*e)*ArcTan[Rt[c*d/e+a*e/d,2]*x/Sqrt[a+c*x^4]]/(2*d*e*Rt[c*d/e+a*e/d,2]) + 
  (B*d+A*e)*(A+B*x^2)*Sqrt[A^2*(a+c*x^4)/(a*(A+B*x^2)^2)]/(4*d*e*A*q*Sqrt[a+c*x^4])*
    EllipticPi[Cancel[-(B*d-A*e)^2/(4*d*e*A*B)],2*ArcTan[q*x],1/2]] /;
FreeQ[{a,c,d,e,A,B},x] && NeQ[c*d^2+a*e^2,0] && NeQ[c*d^2-a*e^2,0] && PosQ[c/a] && EqQ[c*A^2-a*B^2,0]

2:  ∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0 ∧ (c/a)>0 ∧ c (A^2)-a (B^2)≠0

Derivation: Algebraic expansion

Basis: ((A+B (x^2))/(d+e (x^2)))==((B-A q)/(e-d q))-(((B d-A e) (1+q (x^2)))/((e-d q) (d+e (x^2))))

Rule 1.2.2.7.6.1.3.1.3.1.2: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0 ∧ (c/a)>0 ∧ c (A^2)-a (B^2)≠0, let q->(Sqrt[(c/a)]), then

∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  ⟶  
((A (c d+a e q)-a B (e+d q))/(c (d^2)-a (e^2))) ∫(1/(Sqrt[a+b (x^2)+c (x^4)]))dx+((a (B d-A e) (e+d q))/(c (d^2)-a (e^2))) ∫((1+q (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx

Program code:

Int[(A_.+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  With[{q=Rt[c/a,2]},
  (A*(c*d+a*e*q)-a*B*(e+d*q))/(c*d^2-a*e^2)*Int[1/Sqrt[a+b*x^2+c*x^4],x] + 
  a*(B*d-A*e)*(e+d*q)/(c*d^2-a*e^2)*Int[(1+q*x^2)/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x]] /;
FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && NeQ[c*d^2-a*e^2,0] && PosQ[c/a] && NeQ[c*A^2-a*B^2,0]

Int[(A_.+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  With[{q=Rt[c/a,2]},
  (A*(c*d+a*e*q)-a*B*(e+d*q))/(c*d^2-a*e^2)*Int[1/Sqrt[a+c*x^4],x] + 
  a*(B*d-A*e)*(e+d*q)/(c*d^2-a*e^2)*Int[(1+q*x^2)/((d+e*x^2)*Sqrt[a+c*x^4]),x]] /;
FreeQ[{a,c,d,e,A,B},x] && NeQ[c*d^2+a*e^2,0] && NeQ[c*d^2-a*e^2,0] && PosQ[c/a] && NeQ[c*A^2-a*B^2,0]

2:  ∫((A+B (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0 ∧ (c/a)≯0

Derivation: Algebraic expansion

Basis: ((A+B (x^2))/(d+e (x^2)))==(B/e)+((e A-d B)/(e (d+e (x^2))))

Rule 1.2.2.7.6.1.3.1.3.2: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0 ∧ (c/a)≯0, then

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

Program code:

Int[(A_.+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  B/e*Int[1/Sqrt[a+b*x^2+c*x^4],x] + (e*A-d*B)/e*Int[1/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x] /;
FreeQ[{a,b,c,d,e,A,B},x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && NeQ[c*d^2-a*e^2,0] && NegQ[c/a]

Int[(A_.+B_.*x_^2)/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  B/e*Int[1/Sqrt[a+c*x^4],x] + (e*A-d*B)/e*Int[1/((d+e*x^2)*Sqrt[a+c*x^4]),x] /;
FreeQ[{a,c,d,e,A,B},x] && NeQ[c*d^2+a*e^2,0] && NeQ[c*d^2-a*e^2,0] && NegQ[c/a]

2.  ∫((A+B (x^2)+C (x^4))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0

1:  ∫((A+B (x^2)+C (x^4))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)==0

Derivation: Algebraic expansion

Rule 1.2.2.7.6.1.3.2.1: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)==0, then

∫((A+B (x^2)+C (x^4))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  ⟶  -(C/(e^2)) ∫((d-e (x^2))/(Sqrt[a+b (x^2)+c (x^4)]))dx+(1/(e^2)) ∫((C (d^2)+A (e^2)+B (e^2) (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx

Program code:

Int[P4x_/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  With[{A=Coeff[P4x,x,0],B=Coeff[P4x,x,2],C=Coeff[P4x,x,4]},
  -C/e^2*Int[(d-e*x^2)/Sqrt[a+b*x^2+c*x^4],x] + 
  1/e^2*Int[(C*d^2+A*e^2+B*e^2*x^2)/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x]] /;
FreeQ[{a,b,c,d,e},x] && PolyQ[P4x,x^2,2] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && EqQ[c*d^2-a*e^2,0]

Int[P4x_/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  With[{A=Coeff[P4x,x,0],B=Coeff[P4x,x,2],C=Coeff[P4x,x,4]},
  -C/e^2*Int[(d-e*x^2)/Sqrt[a+c*x^4],x] + 
  1/e^2*Int[(C*d^2+A*e^2+B*e^2*x^2)/((d+e*x^2)*Sqrt[a+c*x^4]),x]] /;
FreeQ[{a,c,d,e},x] && PolyQ[P4x,x^2,2] && NeQ[c*d^2+a*e^2,0] && EqQ[c*d^2-a*e^2,0]

2.  ∫((A+B (x^2)+C (x^4))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0

1:  ∫((A+B (x^2)+C (x^4))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0 ∧ (c/a)>0 ∧ (b^2)-4 a c≯0

Derivation: Algebraic expansion

Rule 1.2.2.7.6.1.3.2.2.1: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0 ∧ (c/a)>0 ∧ (b^2)-4 a c≯0, let q->(Sqrt[(c/a)]), then

∫((A+B (x^2)+C (x^4))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  ⟶  -(C/(e q)) ∫((1-q (x^2))/(Sqrt[a+b (x^2)+c (x^4)]))dx+(1/(c e)) ∫((A c e+a C d q+(B c e-C (c d-a e q)) (x^2))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx

Program code:

Int[P4x_/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  With[{q=Rt[c/a,2],A=Coeff[P4x,x,0],B=Coeff[P4x,x,2],C=Coeff[P4x,x,4]},
  -C/(e*q)*Int[(1-q*x^2)/Sqrt[a+b*x^2+c*x^4],x] + 
  1/(c*e)*Int[(A*c*e+a*C*d*q+(B*c*e-C*(c*d-a*e*q))*x^2)/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x]] /;
FreeQ[{a,b,c,d,e},x] && PolyQ[P4x,x^2,2] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && NeQ[c*d^2-a*e^2,0] && PosQ[c/a] && Not[GtQ[b^2-4*a*c,0]]

Int[P4x_/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  With[{q=Rt[c/a,2],A=Coeff[P4x,x,0],B=Coeff[P4x,x,2],C=Coeff[P4x,x,4]},
  -C/(e*q)*Int[(1-q*x^2)/Sqrt[a+c*x^4],x] + 
  1/(c*e)*Int[(A*c*e+a*C*d*q+(B*c*e-C*(c*d-a*e*q))*x^2)/((d+e*x^2)*Sqrt[a+c*x^4]),x]] /;
FreeQ[{a,c,d,e},x] && PolyQ[P4x,x^2,2] && NeQ[c*d^2+a*e^2,0] && NeQ[c*d^2-a*e^2,0] && PosQ[c/a]

2:  ∫((A+B (x^2)+C (x^4))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0

Derivation: Algebraic expansion (polynomial division)

Rule 1.2.2.7.6.1.3.2.2.2: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ c (d^2)-a (e^2)≠0, then

∫((A+B (x^2)+C (x^4))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  ⟶  -(1/(e^2)) ∫((C d-B e-C e (x^2))/(Sqrt[a+b (x^2)+c (x^4)]))dx+((C (d^2)-B d e+A (e^2))/(e^2)) ∫(1/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx

Program code:

Int[P4x_/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  With[{A=Coeff[P4x,x,0],B=Coeff[P4x,x,2],C=Coeff[P4x,x,4]},
  -1/e^2*Int[(C*d-B*e-C*e*x^2)/Sqrt[a+b*x^2+c*x^4],x] + 
  (C*d^2-B*d*e+A*e^2)/e^2*Int[1/((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x]] /;
FreeQ[{a,b,c,d,e},x] && PolyQ[P4x,x^2,2] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && NeQ[c*d^2-a*e^2,0]

Int[P4x_/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  With[{A=Coeff[P4x,x,0],B=Coeff[P4x,x,2],C=Coeff[P4x,x,4]},
  -1/e^2*Int[(C*d-B*e-C*e*x^2)/Sqrt[a+c*x^4],x] + 
  (C*d^2-B*d*e+A*e^2)/e^2*Int[1/((d+e*x^2)*Sqrt[a+c*x^4]),x]] /;
FreeQ[{a,c,d,e},x] && PolyQ[P4x,x^2,2] && NeQ[c*d^2+a*e^2,0] && NeQ[c*d^2-a*e^2,0]

3:  ∫(((Subscript[P, q])[x])/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ q>4

Rule 1.2.2.7.6.1.3.3: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ q>4, then

∫(((Subscript[P, q])[x])/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx  ⟶  
(((Subscript[P, q])[x,q] (x^(q-5)) (Sqrt[a+b (x^2)+c (x^4)]))/(c e (q-3)))+(1/(c e (q-3))) ∫((c e (q-3) (Subscript[P, q])[x]-(Subscript[P, q])[x,q] (x^(q-6)) (d+e (x^2)) (a (q-5)+b (q-4) (x^2)+c (q-3) (x^4)))/((d+e (x^2)) (Sqrt[a+b (x^2)+c (x^4)])))dx

Program code:

Int[Px_/((d_+e_.*x_^2)*Sqrt[a_+b_.*x_^2+c_.*x_^4]),x_Symbol] :=
  With[{q=Expon[Px,x]},
  Coeff[Px,x,q]*x^(q-5)*Sqrt[a+b*x^2+c*x^4]/(c*e*(q-3)) + 
  1/(c*e*(q-3))*
    Int[(c*e*(q-3)*Px-Coeff[Px,x,q]*x^(q-6)*(d+e*x^2)*(a*(q-5)+b*(q-4)*x^2+c*(q-3)*x^4))/
      ((d+e*x^2)*Sqrt[a+b*x^2+c*x^4]),x] /;
 GtQ[q,4]] /;
FreeQ[{a,b,c,d,e},x] && PolyQ[Px,x] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0]

Int[Px_/((d_+e_.*x_^2)*Sqrt[a_+c_.*x_^4]),x_Symbol] :=
  With[{q=Expon[Px,x]},
  Coeff[Px,x,q]*x^(q-5)*Sqrt[a+c*x^4]/(c*e*(q-3)) + 
  1/(c*e*(q-3))*
    Int[(c*e*(q-3)*Px-Coeff[Px,x,q]*x^(q-6)*(d+e*x^2)*(a*(q-5)+c*(q-3)*x^4))/((d+e*x^2)*Sqrt[a+c*x^4]),x] /;
 GtQ[q,4]] /;
FreeQ[{a,c,d,e},x] && PolyQ[Px,x] && NeQ[c*d^2+a*e^2,0]

x:  ∫(((Subscript[P, q])[(x^2)] (((a+b (x^2)+c (x^4)))^p))/(d+e (x^2)))dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ p<-1

Derivation: Algebraic expansion and trinomial recurrence 2b

Rule 1.2.2.7.6.x: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ p<-1, let (Subscript[Q, q-2])[(x^2)]->PolynomialQuotient[(Subscript[P, q])[(x^2)],a+b (x^2)+c (x^4),x] and A+B (x^2)->PolynomialRemainder[(Subscript[P, q])[(x^2)],a+b (x^2)+c (x^4),x], then

∫(((Subscript[P, q])[(x^2)] (((a+b (x^2)+c (x^4)))^p))/(d+e (x^2)))dx  ⟶  

(B/e) ∫(((a+b (x^2)+c (x^4)))^p)dx-((B d-A e)/e)∫((((a+b (x^2)+c (x^4)))^p)/(d+e (x^2)))dx+∫(((Subscript[Q, q-2])[(x^2)] (((a+b (x^2)+c (x^4)))^(p+1)))/(d+e (x^2)))dx  ⟶  

-((B x ((b^2)-2 a c+b c (x^2)) (((a+b (x^2)+c (x^4)))^(p+1)))/(2 a e (p+1) ((b^2)-4 a c)))+(((B d-A e) x ((b^2) c d-2 a (c^2) d-(b^3) e+3 a b c e+c (b c d-(b^2) e+2 a c e) (x^2)) (((a+b (x^2)+c (x^4)))^(p+1)))/(2 a e (p+1) ((b^2)-4 a c) (c (d^2)-b d e+a (e^2))))+
∫((((a+b (x^2)+c (x^4)))^(p+1))/(d+e (x^2))) ((((Subscript[P, q])[(x^2)])/(a+b (x^2)+c (x^4)))-((d+e (x^2))/(((a+b (x^2)+c (x^4)))^(p+1)))·
(∂_x)(-((B x ((b^2)-2 a c+b c (x^2)) (((a+b (x^2)+c (x^4)))^(p+1)))/(2 a e (p+1) ((b^2)-4 a c)))+(((B d-A e) x ((b^2) c d-2 a (c^2) d-(b^3) e+3 a b c e+c (b c d-(b^2) e+2 a c e) (x^2)) (((a+b (x^2)+c (x^4)))^(p+1)))/(2 a e (p+1) ((b^2)-4 a c) (c (d^2)-b d e+a (e^2)))))dx

Program code:

(* Int[Pq_*(a_+b_.*x_^2+c_.*x_^4)^p_/(d_+e_.*x_^2),x_Symbol] :=
  With[{A=Coeff[PolynomialRemainder[Pq,a+b*x^2+c*x^4,x],x,0],
        B=Coeff[PolynomialRemainder[Pq,a+b*x^2+c*x^4,x],x,2]},
  -B*x*(b^2-2*a*c+b*c*x^2)*(a+b*x^2+c*x^4)^(p+1)/(2*a*e*(p+1)*(b^2-4*a*c)) + 
  (B*d-A*e)*x*(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e+c*(b*c*d-b^2*e+2*a*c*e)*x^2)*(a+b*x^2+c*x^4)^(p+1)/
    (2*a*e*(p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)) + 
  Int[(a+b*x^2+c*x^4)^(p+1)/(d+e*x^2)*ExpandToSum[Pq/(a+b*x^2+c*x^4)-(d+e*x^2)/(a+b*x^2+c*x^4)^(p+1)*
    D[-B*x*(b^2-2*a*c+b*c*x^2)*(a+b*x^2+c*x^4)^(p+1)/(2*a*e*(p+1)*(b^2-4*a*c)) + 
      (B*d-A*e)*x*(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*e+c*(b*c*d-b^2*e+2*a*c*e)*x^2)*(a+b*x^2+c*x^4)^(p+1)/
       (2*a*e*(p+1)*(b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)),x],x],x]] /;
FreeQ[{a,b,c,d,e},x] && PolyQ[Pq,x^2] && Expon[Pq,x^2]>0 && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && LtQ[p,-1] *)

2:  ∫P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  when (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ p+(1/2)∈ℤ ∧ q∈ℤ

Derivation: Algebraic expansion

Rule 1.2.2.7.6.2: If  (b^2)-4 a c≠0 ∧ c (d^2)-b d e+a (e^2)≠0 ∧ p+(1/2)∈ℤ ∧ q∈ℤ, then

∫P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  ⟶  ∫(1/(Sqrt[a+b (x^2)+c (x^4)]))ExpandIntegrand[P[(x^2)] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^(p+(1/2))),x] dx

Program code:

Int[Px_*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_,x_Symbol] :=
  Int[ExpandIntegrand[1/Sqrt[a+b*x^2+c*x^4],Px*(d+e*x^2)^q*(a+b*x^2+c*x^4)^(p+1/2),x],x] /;
FreeQ[{a,b,c,d,e},x] && PolyQ[Px,x^2] && NeQ[b^2-4*a*c,0] && NeQ[c*d^2-b*d*e+a*e^2,0] && IntegerQ[p+1/2] && IntegerQ[q]

Int[Px_*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_,x_Symbol] :=
  Int[ExpandIntegrand[1/Sqrt[a+c*x^4],Px*(d+e*x^2)^q*(a+c*x^4)^(p+1/2),x],x] /;
FreeQ[{a,c,d,e},x] && PolyQ[Px,x^2] && NeQ[c*d^2+a*e^2,0] && IntegerQ[p+1/2] && IntegerQ[q]

U:  ∫P[x] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx

Rule 1.2.2.7.U:

∫P[x] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx  ⟶  ∫P[x] (((d+e (x^2)))^q) (((a+b (x^2)+c (x^4)))^p)dx

Program code:

Int[Px_*(d_+e_.*x_^2)^q_.*(a_+b_.*x_^2+c_.*x_^4)^p_.,x_Symbol] :=
  Unintegrable[Px*(d+e*x^2)^q*(a+b*x^2+c*x^4)^p,x] /;
FreeQ[{a,b,c,d,e,p,q},x] && PolyQ[Px,x]

Int[Px_*(d_+e_.*x_^2)^q_.*(a_+c_.*x_^4)^p_.,x_Symbol] :=
  Unintegrable[Px*(d+e*x^2)^q*(a+c*x^4)^p,x] /;
FreeQ[{a,c,d,e,p,q},x] && PolyQ[Px,x]