\(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x)^3 (c i+d i x)^2} \, dx\) [149]

Optimal result

Integrand size = 43, antiderivative size = 361 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\frac {B d^3 n (a+b x)}{(b c-a d)^4 g^3 i^2 (c+d x)}-\frac {b B n (c+d x)^2 \left (b-\frac {6 d (a+b x)}{c+d x}\right )^2}{4 (b c-a d)^4 g^3 i^2 (a+b x)^2}-\frac {d^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^3 i^2 (c+d x)}+\frac {3 b^2 d (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^3 i^2 (a+b x)}-\frac {b^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^3 i^2 (a+b x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^4 g^3 i^2}-\frac {3 b B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^3 i^2} \] Output:

B*d^3*n*(b*x+a)/(-a*d+b*c)^4/g^3/i^2/(d*x+c)-1/4*b*B*n*(d*x+c)^2*(b-6*d*(b 
*x+a)/(d*x+c))^2/(-a*d+b*c)^4/g^3/i^2/(b*x+a)^2-d^3*(b*x+a)*(A+B*ln(e*((b* 
x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^3/i^2/(d*x+c)+3*b^2*d*(d*x+c)*(A+B*ln(e*( 
(b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^3/i^2/(b*x+a)-1/2*b^3*(d*x+c)^2*(A+B*l 
n(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^3/i^2/(b*x+a)^2+3*b*d^2*(A+B*ln(e 
*((b*x+a)/(d*x+c))^n))*ln((b*x+a)/(d*x+c))/(-a*d+b*c)^4/g^3/i^2-3/2*b*B*d^ 
2*n*ln((b*x+a)/(d*x+c))^2/(-a*d+b*c)^4/g^3/i^2
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.69 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.32 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\frac {-\frac {b B (b c-a d)^2 n}{(a+b x)^2}+\frac {8 b^2 B c d n}{a+b x}-\frac {8 a b B d^2 n}{a+b x}+\frac {2 b B d (b c-a d) n}{a+b x}-\frac {4 b B c d^2 n}{c+d x}+\frac {4 a B d^3 n}{c+d x}+6 b B d^2 n \log (a+b x)-\frac {2 b (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2}+\frac {8 b d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}+\frac {4 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x}+12 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 b B d^2 n \log (c+d x)-12 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-6 b B d^2 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+6 b B d^2 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{4 (b c-a d)^4 g^3 i^2} \] Input:

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^3*(c*i + d 
*i*x)^2),x]
 

Output:

(-((b*B*(b*c - a*d)^2*n)/(a + b*x)^2) + (8*b^2*B*c*d*n)/(a + b*x) - (8*a*b 
*B*d^2*n)/(a + b*x) + (2*b*B*d*(b*c - a*d)*n)/(a + b*x) - (4*b*B*c*d^2*n)/ 
(c + d*x) + (4*a*B*d^3*n)/(c + d*x) + 6*b*B*d^2*n*Log[a + b*x] - (2*b*(b*c 
 - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 + (8*b*d*(b* 
c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) + (4*d^2*(b*c - 
 a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x) + 12*b*d^2*Log[a + 
 b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 6*b*B*d^2*n*Log[c + d*x] - 
12*b*d^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 6*b*B*d^2*n 
*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyL 
og[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 6*b*B*d^2*n*((2*Log[(d*(a + b*x))/( 
-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/( 
b*c - a*d)]))/(4*(b*c - a*d)^4*g^3*i^2)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.74, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2961, 2772, 27, 2010, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{(a g+b g x)^3 (c i+d i x)^2} \, dx\)

\(\Big \downarrow \) 2961

\(\displaystyle \frac {\int \frac {(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^3}d\frac {a+b x}{c+d x}}{g^3 i^2 (b c-a d)^4}\)

\(\Big \downarrow \) 2772

\(\displaystyle \frac {-B n \int -\frac {(c+d x)^3 \left (b^3-\frac {6 d (a+b x) b^2}{c+d x}-\frac {6 d^2 (a+b x)^2 \log \left (\frac {a+b x}{c+d x}\right ) b}{(c+d x)^2}+\frac {2 d^3 (a+b x)^3}{(c+d x)^3}\right )}{2 (a+b x)^3}d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+\frac {3 b^2 d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {d^3 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}+3 b d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (b c-a d)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{2} B n \int \frac {(c+d x)^3 \left (b^3-\frac {6 d (a+b x) b^2}{c+d x}-\frac {6 d^2 (a+b x)^2 \log \left (\frac {a+b x}{c+d x}\right ) b}{(c+d x)^2}+\frac {2 d^3 (a+b x)^3}{(c+d x)^3}\right )}{(a+b x)^3}d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+\frac {3 b^2 d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {d^3 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}+3 b d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (b c-a d)^4}\)

\(\Big \downarrow \) 2010

\(\displaystyle \frac {\frac {1}{2} B n \int \left (\frac {(c+d x)^3 \left (b^3-\frac {6 d (a+b x) b^2}{c+d x}+\frac {2 d^3 (a+b x)^3}{(c+d x)^3}\right )}{(a+b x)^3}-\frac {6 b d^2 (c+d x) \log \left (\frac {a+b x}{c+d x}\right )}{a+b x}\right )d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+\frac {3 b^2 d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {d^3 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}+3 b d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (b c-a d)^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^3 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+\frac {3 b^2 d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {d^3 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}+3 b d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {1}{2} B n \left (-\frac {b^3 (c+d x)^2}{2 (a+b x)^2}+\frac {6 b^2 d (c+d x)}{a+b x}+\frac {2 d^3 (a+b x)}{c+d x}-3 b d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )\right )}{g^3 i^2 (b c-a d)^4}\)

Input:

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^3*(c*i + d*i*x)^ 
2),x]
 

Output:

(-((d^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x)) + (3* 
b^2*d*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) - (b^3*( 
c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + b*x)^2) + 3*b*d 
^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)] + (B*n* 
((2*d^3*(a + b*x))/(c + d*x) + (6*b^2*d*(c + d*x))/(a + b*x) - (b^3*(c + d 
*x)^2)/(2*(a + b*x)^2) - 3*b*d^2*Log[(a + b*x)/(c + d*x)]^2))/2)/((b*c - a 
*d)^4*g^3*i^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

rule 2772

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ 
.))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + 
 b*Log[c*x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] 
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q 
, 1] && EqQ[m, -1])
 

rule 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(976\) vs. \(2(355)=710\).

Time = 30.93 (sec) , antiderivative size = 977, normalized size of antiderivative = 2.71

method	result	size
parallelrisch	\(\frac {6 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{7} d^{6} n +12 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a \,b^{6} d^{6}+6 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{7} c \,d^{5}-6 B \,x^{2} a \,b^{6} d^{6} n^{2}+6 B \,x^{2} b^{7} c \,d^{5} n^{2}+24 A \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{6} d^{6}+12 A \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{7} c \,d^{5}-12 A \,x^{2} a \,b^{6} d^{6} n +12 A \,x^{2} b^{7} c \,d^{5} n +6 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{2} b^{5} d^{6}-3 B x \,a^{2} b^{5} d^{6} n^{2}+9 B x \,b^{7} c^{2} d^{4} n^{2}+12 A x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{5} d^{6}-18 A x \,a^{2} b^{5} d^{6} n +6 A x \,b^{7} c^{2} d^{4} n +6 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{2} b^{5} c \,d^{5}-4 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{3} b^{4} d^{6} n -2 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{7} c^{3} d^{3} n +18 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{7} c \,d^{5} n +12 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a \,b^{6} c \,d^{5}-12 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{5} d^{6} n +6 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{7} c^{2} d^{4} n -6 B x a \,b^{6} c \,d^{5} n^{2}+24 A x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{6} c \,d^{5}+12 A x a \,b^{6} c \,d^{5} n +12 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{6} c^{2} d^{4} n +24 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{6} c \,d^{5} n +6 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{7} d^{6}+12 A \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{7} d^{6}+4 B \,a^{3} b^{4} d^{6} n^{2}-B \,b^{7} c^{3} d^{3} n^{2}-4 A \,a^{3} b^{4} d^{6} n -2 A \,b^{7} c^{3} d^{3} n -6 A \,a^{2} b^{5} c \,d^{5} n +12 A a \,b^{6} c^{2} d^{4} n +12 A \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{5} c \,d^{5}-15 B \,a^{2} b^{5} c \,d^{5} n^{2}+12 B a \,b^{6} c^{2} d^{4} n^{2}}{4 i^{2} g^{3} \left (d x +c \right ) \left (b x +a \right )^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) n \left (d a -b c \right ) b^{4} d^{3}}\)	\(977\)

Input:

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^3/(d*i*x+c*i)^2,x,method=_ 
RETURNVERBOSE)
 

Output:

1/4*(6*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^7*d^6*n+12*B*x^2*ln(e*((b*x+a)/(d 
*x+c))^n)^2*a*b^6*d^6+6*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^7*c*d^5-6*B*x^ 
2*a*b^6*d^6*n^2+6*B*x^2*b^7*c*d^5*n^2+24*A*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a 
*b^6*d^6+12*A*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^7*c*d^5-12*A*x^2*a*b^6*d^6*n 
+12*A*x^2*b^7*c*d^5*n+6*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^5*d^6-3*B*x* 
a^2*b^5*d^6*n^2+9*B*x*b^7*c^2*d^4*n^2+12*A*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2 
*b^5*d^6-18*A*x*a^2*b^5*d^6*n+6*A*x*b^7*c^2*d^4*n+6*B*ln(e*((b*x+a)/(d*x+c 
))^n)^2*a^2*b^5*c*d^5-4*B*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^4*d^6*n-2*B*ln(e 
*((b*x+a)/(d*x+c))^n)*b^7*c^3*d^3*n+18*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^7 
*c*d^5*n+12*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b^6*c*d^5-12*B*x*ln(e*((b*x+ 
a)/(d*x+c))^n)*a^2*b^5*d^6*n+6*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^7*c^2*d^4*n 
-6*B*x*a*b^6*c*d^5*n^2+24*A*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^6*c*d^5+12*A*x 
*a*b^6*c*d^5*n+12*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^6*c^2*d^4*n+24*B*x*ln(e* 
((b*x+a)/(d*x+c))^n)*a*b^6*c*d^5*n+6*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)^2*b^7 
*d^6+12*A*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^7*d^6+4*B*a^3*b^4*d^6*n^2-B*b^7* 
c^3*d^3*n^2-4*A*a^3*b^4*d^6*n-2*A*b^7*c^3*d^3*n-6*A*a^2*b^5*c*d^5*n+12*A*a 
*b^6*c^2*d^4*n+12*A*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^5*c*d^5-15*B*a^2*b^5*c 
*d^5*n^2+12*B*a*b^6*c^2*d^4*n^2)/i^2/g^3/(d*x+c)/(b*x+a)^2/(a^3*d^3-3*a^2* 
b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/n/(a*d-b*c)/b^4/d^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 946 vs. \(2 (355) = 710\).

Time = 0.11 (sec) , antiderivative size = 946, normalized size of antiderivative = 2.62 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^3/(d*i*x+c*i)^2,x, 
algorithm="fricas")
 

Output:

-1/4*(2*A*b^3*c^3 - 12*A*a*b^2*c^2*d + 6*A*a^2*b*c*d^2 + 4*A*a^3*d^3 - 6*( 
2*A*b^3*c*d^2 - 2*A*a*b^2*d^3 + (B*b^3*c*d^2 - B*a*b^2*d^3)*n)*x^2 - 6*(B* 
b^3*d^3*n*x^3 + B*a^2*b*c*d^2*n + (B*b^3*c*d^2 + 2*B*a*b^2*d^3)*n*x^2 + (2 
*B*a*b^2*c*d^2 + B*a^2*b*d^3)*n*x)*log((b*x + a)/(d*x + c))^2 + (B*b^3*c^3 
 - 12*B*a*b^2*c^2*d + 15*B*a^2*b*c*d^2 - 4*B*a^3*d^3)*n - 3*(2*A*b^3*c^2*d 
 + 4*A*a*b^2*c*d^2 - 6*A*a^2*b*d^3 + (3*B*b^3*c^2*d - 2*B*a*b^2*c*d^2 - B* 
a^2*b*d^3)*n)*x + 2*(B*b^3*c^3 - 6*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2 + 2*B*a 
^3*d^3 - 6*(B*b^3*c*d^2 - B*a*b^2*d^3)*x^2 - 3*(B*b^3*c^2*d + 2*B*a*b^2*c* 
d^2 - 3*B*a^2*b*d^3)*x - 6*(B*b^3*d^3*x^3 + B*a^2*b*c*d^2 + (B*b^3*c*d^2 + 
 2*B*a*b^2*d^3)*x^2 + (2*B*a*b^2*c*d^2 + B*a^2*b*d^3)*x)*log((b*x + a)/(d* 
x + c)))*log(e) - 2*(6*A*a^2*b*c*d^2 + 3*(B*b^3*d^3*n + 2*A*b^3*d^3)*x^3 + 
 3*(3*B*b^3*c*d^2*n + 2*A*b^3*c*d^2 + 4*A*a*b^2*d^3)*x^2 - (B*b^3*c^3 - 6* 
B*a*b^2*c^2*d + 2*B*a^3*d^3)*n + 3*(4*A*a*b^2*c*d^2 + 2*A*a^2*b*d^3 + (B*b 
^3*c^2*d + 4*B*a*b^2*c*d^2 - 2*B*a^2*b*d^3)*n)*x)*log((b*x + a)/(d*x + c)) 
)/((b^6*c^4*d - 4*a*b^5*c^3*d^2 + 6*a^2*b^4*c^2*d^3 - 4*a^3*b^3*c*d^4 + a^ 
4*b^2*d^5)*g^3*i^2*x^3 + (b^6*c^5 - 2*a*b^5*c^4*d - 2*a^2*b^4*c^3*d^2 + 8* 
a^3*b^3*c^2*d^3 - 7*a^4*b^2*c*d^4 + 2*a^5*b*d^5)*g^3*i^2*x^2 + (2*a*b^5*c^ 
5 - 7*a^2*b^4*c^4*d + 8*a^3*b^3*c^3*d^2 - 2*a^4*b^2*c^2*d^3 - 2*a^5*b*c*d^ 
4 + a^6*d^5)*g^3*i^2*x + (a^2*b^4*c^5 - 4*a^3*b^3*c^4*d + 6*a^4*b^2*c^3*d^ 
2 - 4*a^5*b*c^2*d^3 + a^6*c*d^4)*g^3*i^2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**3/(d*i*x+c*i)**2,x 
)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1724 vs. \(2 (355) = 710\).

Time = 0.15 (sec) , antiderivative size = 1724, normalized size of antiderivative = 4.78 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^3/(d*i*x+c*i)^2,x, 
algorithm="maxima")
 

Output:

1/2*B*((6*b^2*d^2*x^2 - b^2*c^2 + 5*a*b*c*d + 2*a^2*d^2 + 3*(b^2*c*d + 3*a 
*b*d^2)*x)/((b^5*c^3*d - 3*a*b^4*c^2*d^2 + 3*a^2*b^3*c*d^3 - a^3*b^2*d^4)* 
g^3*i^2*x^3 + (b^5*c^4 - a*b^4*c^3*d - 3*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d^3 
 - 2*a^4*b*d^4)*g^3*i^2*x^2 + (2*a*b^4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3*b^2*c 
^2*d^2 + a^4*b*c*d^3 - a^5*d^4)*g^3*i^2*x + (a^2*b^3*c^4 - 3*a^3*b^2*c^3*d 
 + 3*a^4*b*c^2*d^2 - a^5*c*d^3)*g^3*i^2) + 6*b*d^2*log(b*x + a)/((b^4*c^4 
- 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^3*i^2) - 
6*b*d^2*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3 
*b*c*d^3 + a^4*d^4)*g^3*i^2))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - 1/4 
*(b^3*c^3 - 12*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3 - 6*(b^3*c*d^2 - a 
*b^2*d^3)*x^2 + 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x 
^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(b*x + a)^2 + 6*(b^3*d^3*x^3 + a^2* 
b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*l 
og(d*x + c)^2 - 3*(3*b^3*c^2*d - 2*a*b^2*c*d^2 - a^2*b*d^3)*x - 6*(b^3*d^3 
*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2* 
b*d^3)*x)*log(b*x + a) + 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b 
^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x - 2*(b^3*d^3*x^3 + a^2*b*c*d^2 
 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(b*x 
+ a))*log(d*x + c))*B*n/(a^2*b^4*c^5*g^3*i^2 - 4*a^3*b^3*c^4*d*g^3*i^2 + 6 
*a^4*b^2*c^3*d^2*g^3*i^2 - 4*a^5*b*c^2*d^3*g^3*i^2 + a^6*c*d^4*g^3*i^2 ...
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 134.38 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.66 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (B b n - \frac {2 \, {\left (b x + a\right )} B d n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{2} b c g^{3} i^{2}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{2} a d g^{3} i^{2}}{{\left (d x + c\right )}^{2}}} + \frac {B b n - \frac {4 \, {\left (b x + a\right )} B d n}{d x + c} + 2 \, B b \log \left (e\right ) - \frac {4 \, {\left (b x + a\right )} B d \log \left (e\right )}{d x + c} + 2 \, A b - \frac {4 \, {\left (b x + a\right )} A d}{d x + c}}{\frac {{\left (b x + a\right )}^{2} b c g^{3} i^{2}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{2} a d g^{3} i^{2}}{{\left (d x + c\right )}^{2}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^3/(d*i*x+c*i)^2,x, 
algorithm="giac")
 

Output:

-1/4*(2*(B*b*n - 2*(b*x + a)*B*d*n/(d*x + c))*log((b*x + a)/(d*x + c))/((b 
*x + a)^2*b*c*g^3*i^2/(d*x + c)^2 - (b*x + a)^2*a*d*g^3*i^2/(d*x + c)^2) + 
 (B*b*n - 4*(b*x + a)*B*d*n/(d*x + c) + 2*B*b*log(e) - 4*(b*x + a)*B*d*log 
(e)/(d*x + c) + 2*A*b - 4*(b*x + a)*A*d/(d*x + c))/((b*x + a)^2*b*c*g^3*i^ 
2/(d*x + c)^2 - (b*x + a)^2*a*d*g^3*i^2/(d*x + c)^2))*(b*c/(b*c - a*d)^2 - 
 a*d/(b*c - a*d)^2)^2
 

Mupad [B] (verification not implemented)

Time = 28.21 (sec) , antiderivative size = 1016, normalized size of antiderivative = 2.81 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

int((A + B*log(e*((a + b*x)/(c + d*x))^n))/((a*g + b*g*x)^3*(c*i + d*i*x)^ 
2),x)
 

Output:

(3*B*b*d^2*log(e*((a + b*x)/(c + d*x))^n)^2)/(2*g^3*i^2*n*(a*d - b*c)^4) - 
 ((4*A*a^2*d^2 - 2*A*b^2*c^2 - 4*B*a^2*d^2*n - B*b^2*c^2*n + 10*A*a*b*c*d 
+ 11*B*a*b*c*d*n)/(2*(a*d - b*c)) + (3*x^2*(2*A*b^2*d^2 + B*b^2*d^2*n))/(a 
*d - b*c) + (3*x*(6*A*a*b*d^2 + 2*A*b^2*c*d + B*a*b*d^2*n + 3*B*b^2*c*d*n) 
)/(2*(a*d - b*c)))/(x*(2*a^4*d^3*g^3*i^2 + 4*a*b^3*c^3*g^3*i^2 - 6*a^2*b^2 
*c^2*d*g^3*i^2) + x^2*(2*b^4*c^3*g^3*i^2 + 4*a^3*b*d^3*g^3*i^2 - 6*a^2*b^2 
*c*d^2*g^3*i^2) + x^3*(2*a^2*b^2*d^3*g^3*i^2 + 2*b^4*c^2*d*g^3*i^2 - 4*a*b 
^3*c*d^2*g^3*i^2) + 2*a^2*b^2*c^3*g^3*i^2 + 2*a^4*c*d^2*g^3*i^2 - 4*a^3*b* 
c^2*d*g^3*i^2) - (b*d^2*atan((b*d^2*(2*A + B*n)*((a^4*d^4*g^3*i^2 - b^4*c^ 
4*g^3*i^2 + 2*a*b^3*c^3*d*g^3*i^2 - 2*a^3*b*c*d^3*g^3*i^2)/(a^3*d^3*g^3*i^ 
2 - b^3*c^3*g^3*i^2 + 3*a*b^2*c^2*d*g^3*i^2 - 3*a^2*b*c*d^2*g^3*i^2) + 2*b 
*d*x)*(a^3*d^3*g^3*i^2 - b^3*c^3*g^3*i^2 + 3*a*b^2*c^2*d*g^3*i^2 - 3*a^2*b 
*c*d^2*g^3*i^2)*3i)/(g^3*i^2*(6*A*b*d^2 + 3*B*b*d^2*n)*(a*d - b*c)^4))*(2* 
A + B*n)*3i)/(g^3*i^2*(a*d - b*c)^4) - log(e*((a + b*x)/(c + d*x))^n)*(((B 
*(2*a*d + b*c))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (3*B*b*d*x)/(2*(a^2* 
d^2 + b^2*c^2 - 2*a*b*c*d)))/(x*(a^2*d*g^3*i^2 + 2*a*b*c*g^3*i^2) + x^2*(b 
^2*c*g^3*i^2 + 2*a*b*d*g^3*i^2) + a^2*c*g^3*i^2 + b^2*d*g^3*i^2*x^3) + (3* 
B*b*d^2*(b*g^3*i^2*n*x^2*(a*d - b*c) + (a*c*g^3*i^2*n*(a*d - b*c))/d + (g^ 
3*i^2*n*x*(a*d + b*c)*(a*d - b*c))/d))/(g^3*i^2*n*(a*d - b*c)^4*(x*(a^2*d* 
g^3*i^2 + 2*a*b*c*g^3*i^2) + x^2*(b^2*c*g^3*i^2 + 2*a*b*d*g^3*i^2) + a^...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1921, normalized size of antiderivative = 5.32 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^3/(d*i*x+c*i)^2,x)
 

Output:

( - 24*log(a + b*x)*a**4*b*c*d**3*n - 24*log(a + b*x)*a**4*b*d**4*n*x - 12 
*log(a + b*x)*a**3*b**2*c**2*d**2*n - 60*log(a + b*x)*a**3*b**2*c*d**3*n*x 
 - 48*log(a + b*x)*a**3*b**2*d**4*n*x**2 - 18*log(a + b*x)*a**2*b**3*c**2* 
d**2*n**2 - 24*log(a + b*x)*a**2*b**3*c**2*d**2*n*x - 18*log(a + b*x)*a**2 
*b**3*c*d**3*n**2*x - 48*log(a + b*x)*a**2*b**3*c*d**3*n*x**2 - 24*log(a + 
 b*x)*a**2*b**3*d**4*n*x**3 - 36*log(a + b*x)*a*b**4*c**2*d**2*n**2*x - 12 
*log(a + b*x)*a*b**4*c**2*d**2*n*x**2 - 36*log(a + b*x)*a*b**4*c*d**3*n**2 
*x**2 - 12*log(a + b*x)*a*b**4*c*d**3*n*x**3 - 18*log(a + b*x)*b**5*c**2*d 
**2*n**2*x**2 - 18*log(a + b*x)*b**5*c*d**3*n**2*x**3 + 24*log(c + d*x)*a* 
*4*b*c*d**3*n + 24*log(c + d*x)*a**4*b*d**4*n*x + 12*log(c + d*x)*a**3*b** 
2*c**2*d**2*n + 60*log(c + d*x)*a**3*b**2*c*d**3*n*x + 48*log(c + d*x)*a** 
3*b**2*d**4*n*x**2 + 18*log(c + d*x)*a**2*b**3*c**2*d**2*n**2 + 24*log(c + 
 d*x)*a**2*b**3*c**2*d**2*n*x + 18*log(c + d*x)*a**2*b**3*c*d**3*n**2*x + 
48*log(c + d*x)*a**2*b**3*c*d**3*n*x**2 + 24*log(c + d*x)*a**2*b**3*d**4*n 
*x**3 + 36*log(c + d*x)*a*b**4*c**2*d**2*n**2*x + 12*log(c + d*x)*a*b**4*c 
**2*d**2*n*x**2 + 36*log(c + d*x)*a*b**4*c*d**3*n**2*x**2 + 12*log(c + d*x 
)*a*b**4*c*d**3*n*x**3 + 18*log(c + d*x)*b**5*c**2*d**2*n**2*x**2 + 18*log 
(c + d*x)*b**5*c*d**3*n**2*x**3 - 12*log(((a + b*x)**n*e)/(c + d*x)**n)**2 
*a**3*b**2*c*d**3 - 12*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**3*b**2*d** 
4*x - 6*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**3*c**2*d**2 - 30*...