\(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x)^2 (c i+d i x)^2} \, dx\) [148]

Optimal result

Integrand size = 43, antiderivative size = 273 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {B d^2 n (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 B n (c+d x)}{(b c-a d)^3 g^2 i^2 (a+b x)}+\frac {d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^2 i^2}+\frac {b B d n \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^2 i^2} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.44 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.25 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\frac {-\frac {b^2 B c n}{a+b x}+\frac {a b B d n}{a+b x}+\frac {b B c d n}{c+d x}-\frac {a B d^2 n}{c+d x}-\frac {b (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}+\frac {d (-b c+a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x}-2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+b B d n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-b B d n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^3 g^2 i^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.70, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2961, 2772, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

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. \(670\) vs. \(2(273)=546\).

Time = 9.76 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.46

method	result	size
parallelrisch	\(\frac {2 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{3} b^{2} c^{4} d n +B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{3} b^{2} c^{4} d +B x \,a^{4} b \,c^{3} d^{2} n^{2}+B x \,a^{3} b^{2} c^{4} d \,n^{2}+2 A x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{4} b \,c^{3} d^{2}+2 A x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{3} b^{2} c^{4} d -A x \,a^{4} b \,c^{3} d^{2} n +A x \,a^{3} b^{2} c^{4} d n -B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{5} c^{3} d^{2} n +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{3} b^{2} c^{5} n +2 A \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{4} b \,c^{4} d -2 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{4} b \,c^{3} d^{2} n -B x \,a^{5} c^{2} d^{3} n^{2}-B x \,a^{2} b^{3} c^{5} n^{2}+A x \,a^{5} c^{2} d^{3} n -A x \,a^{2} b^{3} c^{5} n +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{4} b \,c^{4} d +B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{3} b^{2} c^{3} d^{2}-B \,x^{2} a^{4} b \,c^{2} d^{3} n^{2}+2 B \,x^{2} a^{3} b^{2} c^{3} d^{2} n^{2}-B \,x^{2} a^{2} b^{3} c^{4} d \,n^{2}+2 A \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{3} b^{2} c^{3} d^{2}+A \,x^{2} a^{4} b \,c^{2} d^{3} n -A \,x^{2} a^{2} b^{3} c^{4} d n +B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{4} b \,c^{3} d^{2}}{i^{2} g^{2} \left (d x +c \right ) \left (b x +a \right ) \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) a^{3} c^{3} n}\)	\(671\)

Input:

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

Output:

(2*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2*c^4*d*n+B*x*ln(e*((b*x+a)/(d*x+c) 
)^n)^2*a^3*b^2*c^4*d+B*x*a^4*b*c^3*d^2*n^2+B*x*a^3*b^2*c^4*d*n^2+2*A*x*ln( 
e*((b*x+a)/(d*x+c))^n)*a^4*b*c^3*d^2+2*A*x*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b 
^2*c^4*d-A*x*a^4*b*c^3*d^2*n+A*x*a^3*b^2*c^4*d*n-B*ln(e*((b*x+a)/(d*x+c))^ 
n)*a^5*c^3*d^2*n+B*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2*c^5*n+2*A*ln(e*((b*x+ 
a)/(d*x+c))^n)*a^4*b*c^4*d-2*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b*c^3*d^2*n 
-B*x*a^5*c^2*d^3*n^2-B*x*a^2*b^3*c^5*n^2+A*x*a^5*c^2*d^3*n-A*x*a^2*b^3*c^5 
*n+B*ln(e*((b*x+a)/(d*x+c))^n)^2*a^4*b*c^4*d+B*x^2*ln(e*((b*x+a)/(d*x+c))^ 
n)^2*a^3*b^2*c^3*d^2-B*x^2*a^4*b*c^2*d^3*n^2+2*B*x^2*a^3*b^2*c^3*d^2*n^2-B 
*x^2*a^2*b^3*c^4*d*n^2+2*A*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2*c^3*d^2+A 
*x^2*a^4*b*c^2*d^3*n-A*x^2*a^2*b^3*c^4*d*n+B*x*ln(e*((b*x+a)/(d*x+c))^n)^2 
*a^4*b*c^3*d^2)/i^2/g^2/(d*x+c)/(b*x+a)/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2 
*d-b^3*c^3)/a^3/c^3/n
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.65 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {A b^{2} c^{2} - A a^{2} d^{2} + {\left (B b^{2} d^{2} n x^{2} + B a b c d n + {\left (B b^{2} c d + B a b d^{2}\right )} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (B b^{2} c^{2} - 2 \, B a b c d + B a^{2} d^{2}\right )} n + 2 \, {\left (A b^{2} c d - A a b d^{2}\right )} x + {\left (B b^{2} c^{2} - B a^{2} d^{2} + 2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x + 2 \, {\left (B b^{2} d^{2} x^{2} + B a b c d + {\left (B b^{2} c d + B a b d^{2}\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right ) + {\left (2 \, A b^{2} d^{2} x^{2} + 2 \, A a b c d + {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} n + 2 \, {\left (A b^{2} c d + A a b d^{2} + {\left (B b^{2} c d - B a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} g^{2} i^{2} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} g^{2} i^{2} x + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} g^{2} i^{2}} \] Input:

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

Output:

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

Sympy [F(-1)]

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

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**2/(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. 862 vs. \(2 (273) = 546\).

Time = 0.08 (sec) , antiderivative size = 862, normalized size of antiderivative = 3.16 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 103.27 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.35 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-{\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} {\left (\frac {{\left (d x + c\right )} B n \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )} g^{2} i^{2}} + \frac {{\left (B n + B \log \left (e\right ) + A\right )} {\left (d x + c\right )}}{{\left (b x + a\right )} g^{2} i^{2}}\right )} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 26.37 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.58 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\frac {B\,b\,d\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{g^2\,i^2\,n\,{\left (a\,d-b\,c\right )}^3}-\frac {A\,b\,c}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {A\,a\,d}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}+\frac {B\,a\,d\,n}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {B\,b\,c\,n}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {2\,A\,b\,d\,x}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {B\,a\,d\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {B\,b\,c\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {2\,B\,b\,d\,x\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {A\,b\,d\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,4{}\mathrm {i}}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^3} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 961, normalized size of antiderivative = 3.52 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*log(a + b*x)*a**3*b*c*d**2*n - 2*log(a + b*x)*a**3*b*d**3*n*x - 2*lo 
g(a + b*x)*a**2*b**2*c**2*d*n - 4*log(a + b*x)*a**2*b**2*c*d**2*n*x - 2*lo 
g(a + b*x)*a**2*b**2*d**3*n*x**2 - 2*log(a + b*x)*a*b**3*c**2*d*n*x - 2*lo 
g(a + b*x)*a*b**3*c*d**2*n*x**2 + 2*log(c + d*x)*a**3*b*c*d**2*n + 2*log(c 
 + d*x)*a**3*b*d**3*n*x + 2*log(c + d*x)*a**2*b**2*c**2*d*n + 4*log(c + d* 
x)*a**2*b**2*c*d**2*n*x + 2*log(c + d*x)*a**2*b**2*d**3*n*x**2 + 2*log(c + 
 d*x)*a*b**3*c**2*d*n*x + 2*log(c + d*x)*a*b**3*c*d**2*n*x**2 - log(((a + 
b*x)**n*e)/(c + d*x)**n)**2*a**2*b**2*c*d**2 - log(((a + b*x)**n*e)/(c + d 
*x)**n)**2*a**2*b**2*d**3*x - log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**3 
*c**2*d - 2*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**3*c*d**2*x - log((( 
a + b*x)**n*e)/(c + d*x)**n)**2*a*b**3*d**3*x**2 - log(((a + b*x)**n*e)/(c 
 + d*x)**n)**2*b**4*c**2*d*x - log(((a + b*x)**n*e)/(c + d*x)**n)**2*b**4* 
c*d**2*x**2 + log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b*d**3*n + log(((a + 
 b*x)**n*e)/(c + d*x)**n)*a**2*b**2*c*d**2*n + 2*log(((a + b*x)**n*e)/(c + 
 d*x)**n)*a**2*b**2*d**3*n*x - log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**3*c 
**2*d*n - log(((a + b*x)**n*e)/(c + d*x)**n)*b**4*c**3*n - 2*log(((a + b*x 
)**n*e)/(c + d*x)**n)*b**4*c**2*d*n*x + a**4*d**3*n - a**3*b*c*d**2*n - a* 
*3*b*d**3*n**2 + a**2*b**2*c**2*d*n + a**2*b**2*c*d**2*n**2 - 2*a**2*b**2* 
d**3*n*x**2 - a*b**3*c**3*n + a*b**3*c**2*d*n**2 + 2*a*b**3*c*d**2*n*x**2 
- b**4*c**3*n**2)/(g**2*n*(a**5*c*d**4 + a**5*d**5*x - 2*a**4*b*c**2*d*...