Rules for integrands of the form  (((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)

1:  ∫(((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p) dx  when  b c-a d≠0 ∧ p∈(SuperPlus[ℤ])

Derivation: Integration by parts

Basis: 1==(∂_x)((a+b x)/b)

Basis: (∂_x)(((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)==B n p (b c-a d) ((((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^(p-1))/((a+b x) (c+d x)))

Rule: If  b c-a d≠0 ∧ p∈(SuperPlus[ℤ]), then

∫(((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p) dx  ⟶  (((a+b x) (((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p))/b)-((B n p (b c-a d))/b) ∫((((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^(p-1))/(c+d x))dx

Program code:

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

Int[(A_.+B_.*Log[e_.*(a_.+b_.*x_)^n_.*(c_.+d_.*x_)^mn_])^p_.,x_Symbol] :=
  (a+b*x)*(A+B*Log[e*(a+b*x)^n/(c+d*x)^n])^p/b - 
  B*n*p*(b*c-a*d)/b*Int[(A+B*Log[e*(a+b*x)^n/(c+d*x)^n])^(p-1)/(c+d*x),x] /;
FreeQ[{a,b,c,d,e,A,B,n},x] && EqQ[n+mn,0] && NeQ[b*c-a*d,0] && IGtQ[p,0]

Note: This rule unifies the above two rules, but is inelegant...

(* Int[(A_.+B_.*Log[e_.*((a_.+b_.*x_)^n1_.*(c_.+d_.*x_)^n2_)^n_.])^p_.,x_Symbol] :=
  (a+b*x)*(A+B*Log[e*((a+b*x)^n1/(c+d*x)^n1)^n])^p/b - 
  B*n*p*(b*c-a*d)/b*Int[(A+B*Log[e*((a+b*x)^n1/(c+d*x)^n1)^n])^(p-1)/(c+d*x),x] /;
FreeQ[{a,b,c,d,e,A,B,n},x] && EqQ[n1+n2,0] && GtQ[n1,0] && (EqQ[n1,1] || EqQ[n,1]) && NeQ[b*c-a*d,0] && IGtQ[p,0] *)

U:  ∫(((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx

Rule:

∫(((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx  ⟶  ∫(((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx

Program code:

Int[(A_.+B_.*Log[e_.*((a_.+b_.*x_)/(c_.+d_.*x_))^n_.])^p_,x_Symbol] :=
  Unintegrable[(A+B*Log[e*((a+b*x)/(c+d*x))^n])^p,x] /;
FreeQ[{a,b,c,d,e,A,B,n,p},x]

Int[(A_.+B_.*Log[e_.*(a_.+b_.*x_)^n_.*(c_.+d_.*x_)^mn_])^p_,x_Symbol] :=
  Unintegrable[(A+B*Log[e*(a+b*x)^n/(c+d*x)^n])^p,x] /;
FreeQ[{a,b,c,d,e,A,B,n,p},x] && EqQ[n+mn,0]

N:  ∫(((A+B Log[e ((((u/v)))^n)]))^p)dx  when  u==a+b x ∧ v==c+d x

Derivation: Algebraic normalization

Rule: If  u==a+b x ∧ v==c+d x, then

∫(((A+B Log[e ((((u/v)))^n)]))^p)dx  ⟶  ∫(((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx

Program code:

Int[(A_.+B_.*Log[e_.*(u_/v_)^n_.])^p_.,x_Symbol] :=
  Int[(A+B*Log[e*(ExpandToSum[u,x]/ExpandToSum[v,x])^n])^p,x] /;
FreeQ[{e,A,B,n,p},x] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]]

Int[(A_.+B_.*Log[e_.*u_^n_.*v_^mn_])^p_.,x_Symbol] :=
  Int[(A+B*Log[e*ExpandToSum[u,x]^n/ExpandToSum[v,x]^n])^p,x] /;
FreeQ[{e,A,B,n,p},x] && EqQ[n+mn,0] && IGtQ[n,0] && LinearQ[{u,v},x] && Not[LinearMatchQ[{u,v},x]]

Rules for integrands of the form  (((f+g x))^m) (((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)

1.  ∫(((f+g x))^m) (A+B Log[e (((((a+b x)/(c+d x))))^n)])dx  when  b c-a d≠0

1.  ∫((A+B Log[e (((((a+b x)/(c+d x))))^n)])/(f+g x))dx  when  b c-a d≠0

1:  ∫((A+B Log[e (((((a+b x)/(c+d x))))^n)])/(f+g x))dx  when  b c-a d≠0 ∧ b f-a g==0

Derivation: Integration by parts

Basis: If  b f-a g==0, then (1/(f+g x))==-(∂_x)((Log[-((b c-a d)/(d (a+b x)))])/g)

Basis: (∂_x)(A+B Log[e (((((a+b x)/(c+d x))))^n)])==((B n (b c-a d))/((a+b x) (c+d x)))

Rule: If  b c-a d≠0 ∧ b f-a g==0, then

∫((A+B Log[e (((((a+b x)/(c+d x))))^n)])/(f+g x))dx  ⟶  -((Log[-((b c-a d)/(d (a+b x)))] (A+B Log[e (((((a+b x)/(c+d x))))^n)]))/g)+((B n (b c-a d))/g)∫((Log[-((b c-a d)/(d (a+b x)))])/((a+b x) (c+d x)))dx

Program code:

Int[(A_.+B_.*Log[e_.*((a_.+b_.*x_)/(c_.+d_.*x_))^n_.])/(f_.+g_.*x_),x_Symbol] :=
  -Log[-(b*c-a*d)/(d*(a+b*x))]*(A+B*Log[e*((a+b*x)/(c+d*x))^n])/g + 
  B*n*(b*c-a*d)/g*Int[Log[-(b*c-a*d)/(d*(a+b*x))]/((a+b*x)*(c+d*x)),x] /;
FreeQ[{a,b,c,d,e,f,g,A,B,n},x] && NeQ[b*c-a*d,0] && EqQ[b*f-a*g,0]

Int[(A_.+B_.*Log[e_.*(a_.+b_.*x_)^n_.*(c_.+d_.*x_)^mn_])/(f_.+g_.*x_),x_Symbol] :=
  -Log[-(b*c-a*d)/(d*(a+b*x))]*(A+B*Log[e*(a+b*x)^n/(c+d*x)^n])/g + 
  B*n*(b*c-a*d)/g*Int[Log[-(b*c-a*d)/(d*(a+b*x))]/((a+b*x)*(c+d*x)),x] /;
FreeQ[{a,b,c,d,e,f,g,A,B,n},x] && EqQ[n+mn,0] && NeQ[b*c-a*d,0] && EqQ[b*f-a*g,0]

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

Derivation: Integration by parts

Basis: If  d f-c g==0, then (1/(f+g x))==-(∂_x)((Log[((b c-a d)/(b (c+d x)))])/g)

Basis: (∂_x)(A+B Log[e (((((a+b x)/(c+d x))))^n)])==((B n (b c-a d))/((a+b x) (c+d x)))

Rule: If  b c-a d≠0 ∧ d f-c g==0, then

∫((A+B Log[e (((((a+b x)/(c+d x))))^n)])/(f+g x))dx  ⟶  -((Log[((b c-a d)/(b (c+d x)))] (A+B Log[e (((((a+b x)/(c+d x))))^n)]))/g)+((B n (b c-a d))/g)∫((Log[((b c-a d)/(b (c+d x)))])/((a+b x) (c+d x)))dx

Program code:

Int[(A_.+B_.*Log[e_.*((a_.+b_.*x_)/(c_.+d_.*x_))^n_.])/(f_.+g_.*x_),x_Symbol] :=
  -Log[(b*c-a*d)/(b*(c+d*x))]*(A+B*Log[e*((a+b*x)/(c+d*x))^n])/g + 
  B*n*(b*c-a*d)/g*Int[Log[(b*c-a*d)/(b*(c+d*x))]/((a+b*x)*(c+d*x)),x] /;
FreeQ[{a,b,c,d,e,f,g,A,B,n},x] && NeQ[b*c-a*d,0] && EqQ[d*f-c*g,0]

Int[(A_.+B_.*Log[e_.*(a_.+b_.*x_)^n_.*(c_.+d_.*x_)^mn_])/(f_.+g_.*x_),x_Symbol] :=
  -Log[(b*c-a*d)/(b*(c+d*x))]*(A+B*Log[e*(a+b*x)^n/(c+d*x)^n])/g + 
  B*n*(b*c-a*d)/g*Int[Log[(b*c-a*d)/(b*(c+d*x))]/((a+b*x)*(c+d*x)),x] /;
FreeQ[{a,b,c,d,e,f,g,A,B,n},x] && EqQ[n+mn,0] && NeQ[b*c-a*d,0] && EqQ[d*f-c*g,0]

3:  ∫((A+B Log[e (((((a+b x)/(c+d x))))^n)])/(f+g x))dx  when  b c-a d≠0

Derivation: Integration by parts

Basis: (∂_x)(A+B Log[e (((((a+b x)/(c+d x))))^n)])==((b B n)/(a+b x))-((B d n)/(c+d x))

Rule: If  b c-a d≠0, then

∫((A+B Log[e (((((a+b x)/(c+d x))))^n)])/(f+g x))dx  ⟶  ((Log[f+g x] (A+B Log[e (((((a+b x)/(c+d x))))^n)]))/g)-((b B n)/g)∫((Log[f+g x])/(a+b x))dx+((B d n)/g)∫((Log[f+g x])/(c+d x))dx

Program code:

Int[(A_.+B_.*Log[e_.*((a_.+b_.*x_)/(c_.+d_.*x_))^n_.])/(f_.+g_.*x_),x_Symbol] :=
  Log[f+g*x]*(A+B*Log[e*((a+b*x)/(c+d*x))^n])/g - 
  b*B*n/g*Int[Log[f+g*x]/(a+b*x),x] + 
  B*d*n/g*Int[Log[f+g*x]/(c+d*x),x] /;
FreeQ[{a,b,c,d,e,f,g,A,B,n},x] && NeQ[b*c-a*d,0]

Int[(A_.+B_.*Log[e_.*(a_.+b_.*x_)^n_.*(c_.+d_.*x_)^mn_])/(f_.+g_.*x_),x_Symbol] :=
  Log[f+g*x]*(A+B*Log[e*(a+b*x)^n/(c+d*x)^n])/g - 
  b*B*n/g*Int[Log[f+g*x]/(a+b*x),x] + 
  B*d*n/g*Int[Log[f+g*x]/(c+d*x),x] /;
FreeQ[{a,b,c,d,e,f,g,A,B,n},x] && EqQ[n+mn,0] && NeQ[b*c-a*d,0]

2:  ∫(((f+g x))^m) (A+B Log[e (((((a+b x)/(c+d x))))^n)])dx  when  b c-a d≠0 ∧ m≠-1 ∧ m≠-2

Derivation: Integration by parts

Basis: (∂_x)(A+B Log[e (((((a+b x)/(c+d x))))^n)])==((B n (b c-a d))/((a+b x) (c+d x)))

Rule: If  b c-a d≠0 ∧ m≠-1 ∧ m≠-2, then

∫(((f+g x))^m) (A+B Log[e (((((a+b x)/(c+d x))))^n)])dx  ⟶  (((((f+g x))^(m+1)) (A+B Log[e (((((a+b x)/(c+d x))))^n)]))/(g (m+1)))-((B n (b c-a d))/(g (m+1))) ∫((((f+g x))^(m+1))/((a+b x) (c+d x)))dx

Program code:

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

Int[(f_.+g_.*x_)^m_.*(A_.+B_.*Log[e_.*(a_.+b_.*x_)^n_.*(c_.+d_.*x_)^mn_]),x_Symbol] :=
  (f+g*x)^(m+1)*(A+B*Log[e*(a+b*x)^n/(c+d*x)^n])/(g*(m+1)) - 
  B*n*(b*c-a*d)/(g*(m+1))*Int[(f+g*x)^(m+1)/((a+b*x)*(c+d*x)),x] /;
FreeQ[{a,b,c,d,e,f,g,A,B,m,n},x] && EqQ[n+mn,0] && NeQ[b*c-a*d,0] && NeQ[m,-1] && Not[EqQ[m,-2] && IntegerQ[n]]

2.  ∫(((f+g x))^m) (((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx  when  b c-a d≠0 ∧ (m|p)∈ℤ

1:  ∫(((f+g x))^m) (((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx  when  b c-a d≠0 ∧ (m|p)∈ℤ ∧ b f-a g==0 ∧ (p>0∨m<-1)

Derivation: Integration by substitution

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

Rule: If  b c-a d≠0 ∧ (m|p)∈ℤ ∧ b f-a g==0 ∧ (p>0∨m<-1), then

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

Program code:

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

Int[(f_.+g_.*x_)^m_.*(A_.+B_.*Log[e_.*(a_.+b_.*x_)^n_.*(c_.+d_.*x_)^mn_])^p_.,x_Symbol] :=
  (b*c-a*d)^(m+1)*(g/b)^m*Subst[Int[x^m*(A+B*Log[e*x^n])^p/(b-d*x)^(m+2),x],x,(a+b*x)/(c+d*x)] /;
FreeQ[{a,b,c,d,e,f,g,A,B,n},x] && EqQ[n+mn,0] && IGtQ[n,0] && NeQ[b*c-a*d,0] && IntegersQ[m,p] && EqQ[b*f-a*g,0] && (GtQ[p,0] || LtQ[m,-1])

2:  ∫(((f+g x))^m) (((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx  when  b c-a d≠0 ∧ (m|p)∈ℤ ∧ d f-c g==0 ∧ (p>0∨m<-1)

Derivation: Integration by substitution

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

Rule: If  b c-a d≠0 ∧ (m|p)∈ℤ ∧ d f-c g==0 ∧ (p>0∨m<-1), then

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

Program code:

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

Int[(f_.+g_.*x_)^m_.*(A_.+B_.*Log[e_.*(a_.+b_.*x_)^n_.*(c_.+d_.*x_)^mn_])^p_.,x_Symbol] :=
  (b*c-a*d)^(m+1)*(g/d)^m*Subst[Int[(A+B*Log[e*x^n])^p/(b-d*x)^(m+2),x],x,(a+b*x)/(c+d*x)] /;
FreeQ[{a,b,c,d,e,f,g,A,B,n},x] && EqQ[n+mn,0] && IGtQ[n,0] && NeQ[b*c-a*d,0] && IntegersQ[m,p] && EqQ[d*f-c*g,0] && (GtQ[p,0] || LtQ[m,-1])

3:  ∫(((f+g x))^m) (((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx  when  b c-a d≠0 ∧ m∈ℤ ∧ p∈(SuperPlus[ℤ])

Derivation: Integration by substitution

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

Rule: If  b c-a d≠0 ∧ m∈ℤ ∧ p∈(SuperPlus[ℤ]), then

∫(((f+g x))^m) (((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx  ⟶  (b c-a d) Subst[∫(((((b f-a g-(d f-c g) x))^m) (((A+B Log[e (x^n)]))^p))/(((b-d x))^(m+2)))dx,x,((a+b x)/(c+d x))]

Program code:

Int[(f_.+g_.*x_)^m_.*(A_.+B_.*Log[e_.*((a_.+b_.*x_)/(c_.+d_.*x_))^n_.])^p_.,x_Symbol] :=
  (b*c-a*d)*Subst[Int[(b*f-a*g-(d*f-c*g)*x)^m*(A+B*Log[e*x^n])^p/(b-d*x)^(m+2),x],x,(a+b*x)/(c+d*x)] /;
FreeQ[{a,b,c,d,e,f,g,A,B,n},x] && NeQ[b*c-a*d,0] && IntegerQ[m] && IGtQ[p,0]

Int[(f_.+g_.*x_)^m_.*(A_.+B_.*Log[e_.*(a_.+b_.*x_)^n_.*(c_.+d_.*x_)^mn_])^p_.,x_Symbol] :=
  (b*c-a*d)*Subst[Int[(b*f-a*g-(d*f-c*g)*x)^m*(A+B*Log[e*x^n])^p/(b-d*x)^(m+2),x],x,(a+b*x)/(c+d*x)] /;
FreeQ[{a,b,c,d,e,f,g,A,B,n},x] && EqQ[n+mn,0] && IGtQ[n,0] && NeQ[b*c-a*d,0] && IntegerQ[m] && IGtQ[p,0]

U:  ∫(((f+g x))^m) (((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx

Rule:

∫(((f+g x))^m) (((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx  ⟶  ∫(((f+g x))^m) (((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx

Program code:

Int[(f_.+g_.*x_)^m_.*(A_.+B_.*Log[e_.*((a_.+b_.*x_)/(c_.+d_.*x_))^n_.])^p_.,x_Symbol] :=
  Unintegrable[(f+g*x)^m*(A+B*Log[e*((a+b*x)/(c+d*x))^n])^p,x] /;
FreeQ[{a,b,c,d,e,f,g,A,B,m,n,p},x]

Int[(f_.+g_.*x_)^m_.*(A_.+B_.*Log[e_.*(a_.+b_.*x_)^n_.*(c_.+d_.*x_)^mn_])^p_.,x_Symbol] :=
  Unintegrable[(f+g*x)^m*(A+B*Log[e*(a+b*x)^n/(c+d*x)^n])^p,x] /;
FreeQ[{a,b,c,d,e,f,g,A,B,m,n,p},x] && EqQ[n+mn,0] && IntegerQ[n]

N:  ∫(w^m) (((A+B Log[e ((((u/v)))^n)]))^p)dx  when  u==a+b x ∧ v==c+d x ∧ w==f+g x

Derivation: Algebraic normalization

Rule: If  u==a+b x ∧ v==c+d x ∧ w==f+g x, then

∫(w^m) (((A+B Log[e ((((u/v)))^n)]))^p)dx  ⟶  ∫(((f+g x))^m) (((A+B Log[e (((((a+b x)/(c+d x))))^n)]))^p)dx

Program code:

Int[w_^m_.*(A_.+B_.*Log[e_.*(u_/v_)^n_.])^p_.,x_Symbol] :=
  Int[ExpandToSum[w,x]^m*(A+B*Log[e*(ExpandToSum[u,x]/ExpandToSum[v,x])^n])^p,x] /;
FreeQ[{e,A,B,m,n,p},x] && LinearQ[{u,v,w},x] && Not[LinearMatchQ[{u,v,w},x]]

Int[w_^m_.*(A_.+B_.*Log[e_.*u_^n_.*v_^mn_])^p_.,x_Symbol] :=
  Int[ExpandToSum[w,x]^m*(A+B*Log[e*ExpandToSum[u,x]^n/ExpandToSum[v,x]^n])^p,x] /;
FreeQ[{e,A,B,m,n,p},x] && EqQ[n+mn,0] && IGtQ[n,0] && LinearQ[{u,v,w},x] && Not[LinearMatchQ[{u,v,w},x]]