\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x) (c i+d i x)^3} \, dx\) [206]

Optimal result

Integrand size = 45, antiderivative size = 402 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {B^2 d^2 n^2 (a+b x)^2}{4 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {4 A b B d n (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}-\frac {4 b B^2 d n^2 (a+b x)}{(b c-a d)^3 g i^3 (c+d x)}+\frac {4 b B^2 d n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^3 g i^3 (c+d x)}-\frac {B d^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g i^3 (c+d x)^2}+\frac {d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^3 g i^3 (c+d x)^2}-\frac {2 b d (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g i^3 (c+d x)}+\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^3 g i^3 n} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(971\) vs. \(2(402)=804\).

Time = 0.95 (sec) , antiderivative size = 971, normalized size of antiderivative = 2.42 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*b^2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^3 - (6*B*n*Log[(a + b*x)/(c + d*x) 
]^2*(-2*A*b^2*c^2 + 4*a*b*B*c*d*n - a^2*B*d^2*n - 4*A*b^2*c*d*x + 4*b^2*B* 
c*d*n*x + 2*a*b*B*d^2*n*x - 2*A*b^2*d^2*x^2 + 3*b^2*B*d^2*n*x^2 - 2*b^2*B* 
(c + d*x)^2*Log[e*((a + b*x)/(c + d*x))^n] + 2*b^2*B*n*(c + d*x)^2*Log[(a 
+ b*x)/(c + d*x)]))/(c + d*x)^2 - (6*B*(b*c - a*d)*n*Log[(a + b*x)/(c + d* 
x)]*(-6*A*b*c + 2*a*A*d + 7*b*B*c*n - a*B*d*n - 4*A*b*d*x + 6*b*B*d*n*x + 
2*B*(-3*b*c + a*d - 2*b*d*x)*Log[e*((a + b*x)/(c + d*x))^n] + 2*B*n*(3*b*c 
 - a*d + 2*b*d*x)*Log[(a + b*x)/(c + d*x)]))/(c + d*x)^2 + (3*(b*c - a*d)^ 
2*(2*A^2 - 2*A*B*n + B^2*n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2* 
B*n*(-2*A + B*n)*Log[(a + b*x)/(c + d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d 
*x)]^2 - 2*B*Log[e*((a + b*x)/(c + d*x))^n]*(-2*A + B*n + 2*B*n*Log[(a + b 
*x)/(c + d*x)])))/(c + d*x)^2 + (6*b*(b*c - a*d)*(2*A^2 - 6*A*B*n + 7*B^2* 
n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2*B*n*(-2*A + 3*B*n)*Log[(a 
 + b*x)/(c + d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 - 2*B*Log[e*((a 
+ b*x)/(c + d*x))^n]*(-2*A + 3*B*n + 2*B*n*Log[(a + b*x)/(c + d*x)])))/(c 
+ d*x) + 6*b^2*Log[a + b*x]*(2*A^2 - 6*A*B*n + 7*B^2*n^2 + 2*B^2*Log[e*((a 
 + b*x)/(c + d*x))^n]^2 + 2*B*n*(-2*A + 3*B*n)*Log[(a + b*x)/(c + d*x)] + 
2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 - 2*B*Log[e*((a + b*x)/(c + d*x))^n]* 
(-2*A + 3*B*n + 2*B*n*Log[(a + b*x)/(c + d*x)])) - 6*b^2*(2*A^2 - 6*A*B*n 
+ 7*B^2*n^2 + 2*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2*B*n*(-2*A + 3*...
 

Rubi [A] (warning: unable to verify)

Time = 0.96 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2961, 2788, 2767, 2009, 2788, 2733, 2009, 2739, 15}

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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2788

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

\(\Big \downarrow \) 2767

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2788

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

\(\Big \downarrow \) 2733

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2739

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

\(\Big \downarrow \) 15

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 2009

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

rule 2733

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b 
*Log[c*x^n])^p, x] - Simp[b*n*p   Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; 
 FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
 

rule 2739

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( 
b*n)   Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} 
, x]
 

rule 2767

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( 
q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x 
^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] 
&& IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
 

rule 2788

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.)) 
/(x_), x_Symbol] :> Simp[d   Int[(d + e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x) 
, x], x] + Simp[e   Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /; F 
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]
 

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. \(1235\) vs. \(2(394)=788\).

Time = 13.41 (sec) , antiderivative size = 1236, normalized size of antiderivative = 3.07

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1236\)

Input:

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

Output:

-1/12*(4*B^2*ln(e*((b*x+a)/(d*x+c))^n)^3*a^2*b^2*c^6+12*A^2*ln(e*((b*x+a)/ 
(d*x+c))^n)*a^2*b^2*c^6-18*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^2*c^4 
*d^2*n+42*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^2*c^4*d^2*n^2+12*A*B*x^2 
*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^2*c^4*d^2-48*A*B*x^2*a^3*b*c^3*d^3*n^2+ 
42*A*B*x^2*a^2*b^2*c^4*d^2*n^2-12*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^3*b* 
c^4*d^2*n-24*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^2*c^5*d*n+36*B^2*x*ln 
(e*((b*x+a)/(d*x+c))^n)*a^3*b*c^4*d^2*n^2+48*B^2*x*ln(e*((b*x+a)/(d*x+c))^ 
n)*a^2*b^2*c^5*d*n^2+24*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^2*c^5*d-60 
*A*B*x*a^3*b*c^4*d^2*n^2+48*A*B*x*a^2*b^2*c^5*d*n^2-48*A*B*ln(e*((b*x+a)/( 
d*x+c))^n)*a^3*b*c^5*d*n+8*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^3*a^2*b^2*c^5*d 
+54*B^2*x*a^3*b*c^4*d^2*n^3-48*B^2*x*a^2*b^2*c^5*d*n^3+12*A^2*x^2*ln(e*((b 
*x+a)/(d*x+c))^n)*a^2*b^2*c^4*d^2+24*A^2*x^2*a^3*b*c^3*d^3*n-18*A^2*x^2*a^ 
2*b^2*c^4*d^2*n+12*A*B*x*a^4*c^3*d^3*n^2+4*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^ 
n)^3*a^2*b^2*c^4*d^2+48*B^2*x^2*a^3*b*c^3*d^3*n^3-45*B^2*x^2*a^2*b^2*c^4*d 
^2*n^3+6*A*B*x^2*a^4*c^2*d^4*n^2-24*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^3*b* 
c^5*d*n+48*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b*c^5*d*n^2+24*A^2*x*ln(e*((b 
*x+a)/(d*x+c))^n)*a^2*b^2*c^5*d+36*A^2*x*a^3*b*c^4*d^2*n-24*A^2*x*a^2*b^2* 
c^5*d*n+12*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a^4*c^4*d^2*n-36*A*B*x^2*ln(e*((b 
*x+a)/(d*x+c))^n)*a^2*b^2*c^4*d^2*n-24*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^3 
*b*c^4*d^2*n-48*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^2*c^5*d*n-3*B^2*x...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1076 vs. \(2 (394) = 788\).

Time = 0.11 (sec) , antiderivative size = 1076, normalized size of antiderivative = 2.68 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

1/12*(18*A^2*b^2*c^2 - 24*A^2*a*b*c*d + 6*A^2*a^2*d^2 + 4*(B^2*b^2*d^2*n^2 
*x^2 + 2*B^2*b^2*c*d*n^2*x + B^2*b^2*c^2*n^2)*log((b*x + a)/(d*x + c))^3 + 
 3*(15*B^2*b^2*c^2 - 16*B^2*a*b*c*d + B^2*a^2*d^2)*n^2 + 6*(3*B^2*b^2*c^2 
- 4*B^2*a*b*c*d + B^2*a^2*d^2 + 2*(B^2*b^2*c*d - B^2*a*b*d^2)*x + 2*(B^2*b 
^2*d^2*x^2 + 2*B^2*b^2*c*d*x + B^2*b^2*c^2)*log((b*x + a)/(d*x + c)))*log( 
e)^2 + 6*(2*A*B*b^2*c^2*n - (4*B^2*a*b*c*d - B^2*a^2*d^2)*n^2 - (3*B^2*b^2 
*d^2*n^2 - 2*A*B*b^2*d^2*n)*x^2 + 2*(2*A*B*b^2*c*d*n - (2*B^2*b^2*c*d + B^ 
2*a*b*d^2)*n^2)*x)*log((b*x + a)/(d*x + c))^2 - 6*(7*A*B*b^2*c^2 - 8*A*B*a 
*b*c*d + A*B*a^2*d^2)*n + 6*(2*A^2*b^2*c*d - 2*A^2*a*b*d^2 + 7*(B^2*b^2*c* 
d - B^2*a*b*d^2)*n^2 - 6*(A*B*b^2*c*d - A*B*a*b*d^2)*n)*x + 6*(6*A*B*b^2*c 
^2 - 8*A*B*a*b*c*d + 2*A*B*a^2*d^2 + 2*(B^2*b^2*d^2*n*x^2 + 2*B^2*b^2*c*d* 
n*x + B^2*b^2*c^2*n)*log((b*x + a)/(d*x + c))^2 - (7*B^2*b^2*c^2 - 8*B^2*a 
*b*c*d + B^2*a^2*d^2)*n + 2*(2*A*B*b^2*c*d - 2*A*B*a*b*d^2 - 3*(B^2*b^2*c* 
d - B^2*a*b*d^2)*n)*x + 2*(2*A*B*b^2*c^2 - (3*B^2*b^2*d^2*n - 2*A*B*b^2*d^ 
2)*x^2 - (4*B^2*a*b*c*d - B^2*a^2*d^2)*n + 2*(2*A*B*b^2*c*d - (2*B^2*b^2*c 
*d + B^2*a*b*d^2)*n)*x)*log((b*x + a)/(d*x + c)))*log(e) + 6*(2*A^2*b^2*c^ 
2 + (8*B^2*a*b*c*d - B^2*a^2*d^2)*n^2 + (7*B^2*b^2*d^2*n^2 - 6*A*B*b^2*d^2 
*n + 2*A^2*b^2*d^2)*x^2 - 2*(4*A*B*a*b*c*d - A*B*a^2*d^2)*n + 2*(2*A^2*b^2 
*c*d + (4*B^2*b^2*c*d + 3*B^2*a*b*d^2)*n^2 - 2*(2*A*B*b^2*c*d + A*B*a*b*d^ 
2)*n)*x)*log((b*x + a)/(d*x + c)))/((b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 + 3*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2126 vs. \(2 (394) = 788\).

Time = 0.18 (sec) , antiderivative size = 2126, normalized size of antiderivative = 5.29 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*B^2*((2*b*d*x + 3*b*c - a*d)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*g* 
i^3*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*g*i^3*x + (b^2*c^4 - 2 
*a*b*c^3*d + a^2*c^2*d^2)*g*i^3) + 2*b^2*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*i^3) - 2*b^2*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*i^3))*log(e*(b*x/(d*x + c) + a/( 
d*x + c))^n)^2 + A*B*((2*b*d*x + 3*b*c - a*d)/((b^2*c^2*d^2 - 2*a*b*c*d^3 
+ a^2*d^4)*g*i^3*x^2 + 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*g*i^3*x + 
 (b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*g*i^3) + 2*b^2*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*i^3) - 2*b^2*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*i^3))*log(e*(b*x/(d 
*x + c) + a/(d*x + c))^n) + 1/12*((45*b^2*c^2 - 48*a*b*c*d + 3*a^2*d^2 + 4 
*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a)^3 - 4*(b^2*d^2*x^2 + 2 
*b^2*c*d*x + b^2*c^2)*log(d*x + c)^3 + 18*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2 
*c^2)*log(b*x + a)^2 + 6*(3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 + 2*(b^2 
*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a))*log(d*x + c)^2 + 42*(b^2*c 
*d - a*b*d^2)*x + 42*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a) - 
6*(7*b^2*d^2*x^2 + 14*b^2*c*d*x + 7*b^2*c^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x 
 + b^2*c^2)*log(b*x + a)^2 + 6*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b 
*x + a))*log(d*x + c))*n^2/(b^3*c^5*g*i^3 - 3*a*b^2*c^4*d*g*i^3 + 3*a^2*b* 
c^3*d^2*g*i^3 - a^3*c^2*d^3*g*i^3 + (b^3*c^3*d^2*g*i^3 - 3*a*b^2*c^2*d^...
 

Giac [A] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 747, normalized size of antiderivative = 1.86 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

1/12*(4*B^2*b^2*n^2*log((b*x + a)/(d*x + c))^3/(b^2*c^2*g*i^3 - 2*a*b*c*d* 
g*i^3 + a^2*d^2*g*i^3) - 6*(4*(b*x + a)*B^2*b*d*n^2/((b^2*c^2*g*i^3 - 2*a* 
b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)) - (b*x + a)^2*B^2*d^2*n^2/((b^2*c^ 
2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)^2) - 2*(B^2*b^2*n*log 
(e) + A*B*b^2*n)/(b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3))*log((b 
*x + a)/(d*x + c))^2 - 6*((B^2*d^2*n^2 - 2*B^2*d^2*n*log(e) - 2*A*B*d^2*n) 
*(b*x + a)^2/((b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)^ 
2) - 8*(B^2*b*d*n^2 - B^2*b*d*n*log(e) - A*B*b*d*n)*(b*x + a)/((b^2*c^2*g* 
i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)))*log((b*x + a)/(d*x + c) 
) + 12*(B^2*b^2*log(e)^2 + 2*A*B*b^2*log(e) + A^2*b^2)*log((b*x + a)/(d*x 
+ c))/(b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3) + 3*(B^2*d^2*n^2 - 
 2*B^2*d^2*n*log(e) + 2*B^2*d^2*log(e)^2 - 2*A*B*d^2*n + 4*A*B*d^2*log(e) 
+ 2*A^2*d^2)*(b*x + a)^2/((b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3 
)*(d*x + c)^2) - 24*(2*B^2*b*d*n^2 - 2*B^2*b*d*n*log(e) + B^2*b*d*log(e)^2 
 - 2*A*B*b*d*n + 2*A*B*b*d*log(e) + A^2*b*d)*(b*x + a)/((b^2*c^2*g*i^3 - 2 
*a*b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)))*(b*c/(b*c - a*d)^2 - a*d/(b*c 
- a*d)^2)
 

Mupad [B] (verification not implemented)

Time = 29.96 (sec) , antiderivative size = 1007, normalized size of antiderivative = 2.50 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

log(e*((a + b*x)/(c + d*x))^n)^2*((b^2*(3*B^2*n - 2*A*B))/(2*g*i^3*n*(a*d 
- b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (B^2*b^2*((c*g*i^3*n*(a*d - b*c) 
)/(2*b) - (g*i^3*n*(a*d - b*c)*(a*d - 2*b*c))/(2*b^2) + (d*g*i^3*n*x*(a*d 
- b*c))/b))/(g*i^3*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(c^2*g*i^ 
3 + d^2*g*i^3*x^2 + 2*c*d*g*i^3*x))) - ((2*A^2*a*d - 6*A^2*b*c + B^2*a*d*n 
^2 - 15*B^2*b*c*n^2 - 2*A*B*a*d*n + 14*A*B*b*c*n)/(2*(a*d - b*c)) - (x*(2* 
A^2*b*d + 7*B^2*b*d*n^2 - 6*A*B*b*d*n))/(a*d - b*c))/(x^2*(2*a*d^3*g*i^3 - 
 2*b*c*d^2*g*i^3) + x*(4*a*c*d^2*g*i^3 - 4*b*c^2*d*g*i^3) - 2*b*c^3*g*i^3 
+ 2*a*c^2*d*g*i^3) - log(e*((a + b*x)/(c + d*x))^n)*((B^2*n)/(x^2*(a*d^3*g 
*i^3 - b*c*d^2*g*i^3) + x*(2*a*c*d^2*g*i^3 - 2*b*c^2*d*g*i^3) - b*c^3*g*i^ 
3 + a*c^2*d*g*i^3) + (b^2*(3*B^2*n - 2*A*B)*((c*g*i^3*n*(a*d - b*c)^2)/(2* 
b) - (g*i^3*n*(a*d - b*c)^2*(a*d - 2*b*c))/(2*b^2) + (d*g*i^3*n*x*(a*d - b 
*c)^2)/b))/(g*i^3*n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(x^2*(a*d^ 
3*g*i^3 - b*c*d^2*g*i^3) + x*(2*a*c*d^2*g*i^3 - 2*b*c^2*d*g*i^3) - b*c^3*g 
*i^3 + a*c^2*d*g*i^3))) + (b^2*atan((b^2*((a^3*d^3*g*i^3 + b^3*c^3*g*i^3 - 
 a*b^2*c^2*d*g*i^3 - a^2*b*c*d^2*g*i^3)/(a^2*d^2*g*i^3 + b^2*c^2*g*i^3 - 2 
*a*b*c*d*g*i^3) + 2*b*d*x)*(A^2 + (7*B^2*n^2)/2 - 3*A*B*n)*(a^2*d^2*g*i^3 
+ b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3)*2i)/(g*i^3*(a*d - b*c)^3*(2*A^2*b^2 + 7 
*B^2*b^2*n^2 - 6*A*B*b^2*n)))*(A^2 + (7*B^2*n^2)/2 - 3*A*B*n)*2i)/(g*i^3*( 
a*d - b*c)^3) - (B^2*b^2*log(e*((a + b*x)/(c + d*x))^n)^3)/(3*g*i^3*n*(...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1640, normalized size of antiderivative = 4.08 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x) (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*( - 12*log(a + b*x)*a**2*b**2*c**3*n + 12*log(a + b*x)*a**2*b**2*c**2*d 
*n**2 - 24*log(a + b*x)*a**2*b**2*c**2*d*n*x + 24*log(a + b*x)*a**2*b**2*c 
*d**2*n**2*x - 12*log(a + b*x)*a**2*b**2*c*d**2*n*x**2 + 12*log(a + b*x)*a 
**2*b**2*d**3*n**2*x**2 + 24*log(a + b*x)*a*b**3*c**3*n**2 - 18*log(a + b* 
x)*a*b**3*c**2*d*n**3 + 48*log(a + b*x)*a*b**3*c**2*d*n**2*x - 36*log(a + 
b*x)*a*b**3*c*d**2*n**3*x + 24*log(a + b*x)*a*b**3*c*d**2*n**2*x**2 - 18*l 
og(a + b*x)*a*b**3*d**3*n**3*x**2 - 24*log(a + b*x)*b**4*c**3*n**3 - 48*lo 
g(a + b*x)*b**4*c**2*d*n**3*x - 24*log(a + b*x)*b**4*c*d**2*n**3*x**2 + 12 
*log(c + d*x)*a**2*b**2*c**3*n - 12*log(c + d*x)*a**2*b**2*c**2*d*n**2 + 2 
4*log(c + d*x)*a**2*b**2*c**2*d*n*x - 24*log(c + d*x)*a**2*b**2*c*d**2*n** 
2*x + 12*log(c + d*x)*a**2*b**2*c*d**2*n*x**2 - 12*log(c + d*x)*a**2*b**2* 
d**3*n**2*x**2 - 24*log(c + d*x)*a*b**3*c**3*n**2 + 18*log(c + d*x)*a*b**3 
*c**2*d*n**3 - 48*log(c + d*x)*a*b**3*c**2*d*n**2*x + 36*log(c + d*x)*a*b* 
*3*c*d**2*n**3*x - 24*log(c + d*x)*a*b**3*c*d**2*n**2*x**2 + 18*log(c + d* 
x)*a*b**3*d**3*n**3*x**2 + 24*log(c + d*x)*b**4*c**3*n**3 + 48*log(c + d*x 
)*b**4*c**2*d*n**3*x + 24*log(c + d*x)*b**4*c*d**2*n**3*x**2 - 4*log(((a + 
 b*x)**n*e)/(c + d*x)**n)**3*b**4*c**3 - 8*log(((a + b*x)**n*e)/(c + d*x)* 
*n)**3*b**4*c**2*d*x - 4*log(((a + b*x)**n*e)/(c + d*x)**n)**3*b**4*c*d**2 
*x**2 - 6*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**2*c*d**2*n - 12*lo 
g(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**3*c**3 + 24*log(((a + b*x)**n*...