\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx\) [199]

Optimal result

Integrand size = 45, antiderivative size = 392 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {2 A B d^2 n (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}+\frac {2 B^2 d^2 n^2 (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {2 b^2 B^2 n^2 (c+d x)}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 B^2 d^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {2 b^2 B n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (a+b x)}+\frac {d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^3 g^2 i^2 n} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(870\) vs. \(2(392)=784\).

Time = 1.12 (sec) , antiderivative size = 870, normalized size of antiderivative = 2.22 \[ \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)^2 (c i+d i x)^2} \, dx=-\frac {2 b B^2 d n^2 (a+b x) (c+d x) \log ^3\left (\frac {a+b x}{c+d x}\right )+3 B n \log ^2\left (\frac {a+b x}{c+d x}\right ) \left (2 a A b c d+b^2 B c^2 n-a^2 B d^2 n+2 A b^2 c d x+2 a A b d^2 x+2 b^2 B c d n x-2 a b B d^2 n x+2 A b^2 d^2 x^2+2 b B d (a+b x) (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 b B d n (a+b x) (c+d x) \log \left (\frac {a+b x}{c+d x}\right )\right )+6 B (b c-a d) n \log \left (\frac {a+b x}{c+d x}\right ) \left (A b c+a A d+b B c n-a B d n+2 A b d x+B (a d+b (c+2 d x)) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n (b c+a d+2 b d x) \log \left (\frac {a+b x}{c+d x}\right )\right )+6 b d (a+b x) (c+d x) \log (a+b x) \left (A^2+2 B^2 n^2+2 A B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2\right )+3 b (b c-a d) (c+d x) \left (A^2+2 A B n+2 B^2 n^2+B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 B n (A+B n) \log \left (\frac {a+b x}{c+d x}\right )+B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (A+B n-B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )+3 d (b c-a d) (a+b x) \left (A^2-2 A B n+2 B^2 n^2+B^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 B n (-A+B n) \log \left (\frac {a+b x}{c+d x}\right )+B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \left (-A+B n+B n \log \left (\frac {a+b x}{c+d x}\right )\right )\right )-6 b d (a+b x) (c+d x) \left (A^2+2 B^2 n^2+2 A B \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )+B^2 \left (\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log \left (\frac {a+b x}{c+d x}\right )\right )^2\right ) \log (c+d x)}{3 (b c-a d)^3 g^2 i^2 (a+b x) (c+d x)} \] Input:

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

Output:

-1/3*(2*b*B^2*d*n^2*(a + b*x)*(c + d*x)*Log[(a + b*x)/(c + d*x)]^3 + 3*B*n 
*Log[(a + b*x)/(c + d*x)]^2*(2*a*A*b*c*d + b^2*B*c^2*n - a^2*B*d^2*n + 2*A 
*b^2*c*d*x + 2*a*A*b*d^2*x + 2*b^2*B*c*d*n*x - 2*a*b*B*d^2*n*x + 2*A*b^2*d 
^2*x^2 + 2*b*B*d*(a + b*x)*(c + d*x)*Log[e*((a + b*x)/(c + d*x))^n] - 2*b* 
B*d*n*(a + b*x)*(c + d*x)*Log[(a + b*x)/(c + d*x)]) + 6*B*(b*c - a*d)*n*Lo 
g[(a + b*x)/(c + d*x)]*(A*b*c + a*A*d + b*B*c*n - a*B*d*n + 2*A*b*d*x + B* 
(a*d + b*(c + 2*d*x))*Log[e*((a + b*x)/(c + d*x))^n] - B*n*(b*c + a*d + 2* 
b*d*x)*Log[(a + b*x)/(c + d*x)]) + 6*b*d*(a + b*x)*(c + d*x)*Log[a + b*x]* 
(A^2 + 2*B^2*n^2 + 2*A*B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x) 
/(c + d*x)]) + B^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + 
d*x)])^2) + 3*b*(b*c - a*d)*(c + d*x)*(A^2 + 2*A*B*n + 2*B^2*n^2 + B^2*Log 
[e*((a + b*x)/(c + d*x))^n]^2 - 2*B*n*(A + B*n)*Log[(a + b*x)/(c + d*x)] + 
 B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 2*B*Log[e*((a + b*x)/(c + d*x))^n]*( 
A + B*n - B*n*Log[(a + b*x)/(c + d*x)])) + 3*d*(b*c - a*d)*(a + b*x)*(A^2 
- 2*A*B*n + 2*B^2*n^2 + B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2*B*n*(-A + 
 B*n)*Log[(a + b*x)/(c + d*x)] + B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 - 2*B* 
Log[e*((a + b*x)/(c + d*x))^n]*(-A + B*n + B*n*Log[(a + b*x)/(c + d*x)])) 
- 6*b*d*(a + b*x)*(c + d*x)*(A^2 + 2*B^2*n^2 + 2*A*B*(Log[e*((a + b*x)/(c 
+ d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) + B^2*(Log[e*((a + b*x)/(c + d*x) 
)^n] - n*Log[(a + b*x)/(c + d*x)])^2)*Log[c + d*x])/((b*c - a*d)^3*g^2*...
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2961, 2795, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((-2*A*B*d^2*n*(a + b*x))/(c + d*x) + (2*B^2*d^2*n^2*(a + b*x))/(c + d*x) 
- (2*b^2*B^2*n^2*(c + d*x))/(a + b*x) - (2*B^2*d^2*n*(a + b*x)*Log[e*((a + 
 b*x)/(c + d*x))^n])/(c + d*x) - (2*b^2*B*n*(c + d*x)*(A + B*Log[e*((a + b 
*x)/(c + d*x))^n]))/(a + b*x) + (d^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c 
+ d*x))^n])^2)/(c + d*x) - (b^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x 
))^n])^2)/(a + b*x) - (2*b*d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3* 
B*n))/((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 2795

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

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. \(1306\) vs. \(2(390)=780\).

Time = 9.77 (sec) , antiderivative size = 1307, normalized size of antiderivative = 3.33

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1307\)

Input:

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

Output:

1/3*(-12*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b*c^3*d^2*n+12*A*B*x*ln(e*((b 
*x+a)/(d*x+c))^n)*a^3*b^2*c^4*d*n+12*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^3 
*b^2*c^3*d^2*n^2+6*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^3*b^2*c^3*d^2-6*A 
*B*x^2*a^4*b*c^2*d^3*n^2+12*A*B*x^2*a^3*b^2*c^3*d^2*n^2-6*A*B*x^2*a^2*b^3* 
c^4*d*n^2-6*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^4*b*c^3*d^2*n+6*B^2*x*ln(e 
*((b*x+a)/(d*x+c))^n)^2*a^3*b^2*c^4*d*n+12*B^2*x*ln(e*((b*x+a)/(d*x+c))^n) 
*a^4*b*c^3*d^2*n^2+12*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2*c^4*d*n^2+6* 
A*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^4*b*c^3*d^2+6*A*B*x*ln(e*((b*x+a)/(d*x 
+c))^n)^2*a^3*b^2*c^4*d+6*A*B*x*a^4*b*c^3*d^2*n^2+6*A*B*x*a^3*b^2*c^4*d*n^ 
2+6*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^5*c^3*d^2*n^2+6*B^2*ln(e*((b*x+a)/(d*x 
+c))^n)*a^3*b^2*c^5*n^2+3*A^2*x*a^5*c^2*d^3*n-3*A^2*x*a^2*b^3*c^5*n+6*A^2* 
ln(e*((b*x+a)/(d*x+c))^n)*a^4*b*c^4*d+6*B^2*x*a^5*c^2*d^3*n^3-6*B^2*x*a^2* 
b^3*c^5*n^3+2*B^2*ln(e*((b*x+a)/(d*x+c))^n)^3*a^4*b*c^4*d-3*B^2*ln(e*((b*x 
+a)/(d*x+c))^n)^2*a^5*c^3*d^2*n+3*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^3*b^2* 
c^5*n+2*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^3*a^3*b^2*c^3*d^2+6*B^2*x^2*a^4* 
b*c^2*d^3*n^3-6*B^2*x^2*a^2*b^3*c^4*d*n^3+2*B^2*x*ln(e*((b*x+a)/(d*x+c))^n 
)^3*a^4*b*c^3*d^2+2*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^3*a^3*b^2*c^4*d-6*B^2* 
x*a^4*b*c^3*d^2*n^3+6*B^2*x*a^3*b^2*c^4*d*n^3+6*A^2*x^2*ln(e*((b*x+a)/(d*x 
+c))^n)*a^3*b^2*c^3*d^2+3*A^2*x^2*a^4*b*c^2*d^3*n-3*A^2*x^2*a^2*b^3*c^4*d* 
n-6*A*B*x*a^5*c^2*d^3*n^2-6*A*B*x*a^2*b^3*c^5*n^2+6*A^2*x*ln(e*((b*x+a)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 983 vs. \(2 (390) = 780\).

Time = 0.11 (sec) , antiderivative size = 983, normalized size of antiderivative = 2.51 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/3*(3*A^2*b^2*c^2 - 3*A^2*a^2*d^2 + 2*(B^2*b^2*d^2*n^2*x^2 + B^2*a*b*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))^3 + 6*( 
B^2*b^2*c^2 - B^2*a^2*d^2)*n^2 + 3*(B^2*b^2*c^2 - 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 + B^2*a*b*c*d + (B^2*b^2*c*d + 
B^2*a*b*d^2)*x)*log((b*x + a)/(d*x + c)))*log(e)^2 + 3*(2*A*B*b^2*d^2*n*x^ 
2 + 2*A*B*a*b*c*d*n + (B^2*b^2*c^2 - B^2*a^2*d^2)*n^2 + 2*((B^2*b^2*c*d - 
B^2*a*b*d^2)*n^2 + (A*B*b^2*c*d + A*B*a*b*d^2)*n)*x)*log((b*x + a)/(d*x + 
c))^2 + 6*(A*B*b^2*c^2 - 2*A*B*a*b*c*d + A*B*a^2*d^2)*n + 6*(A^2*b^2*c*d - 
 A^2*a*b*d^2 + 2*(B^2*b^2*c*d - B^2*a*b*d^2)*n^2)*x + 6*(A*B*b^2*c^2 - A*B 
*a^2*d^2 + (B^2*b^2*d^2*n*x^2 + B^2*a*b*c*d*n + (B^2*b^2*c*d + B^2*a*b*d^2 
)*n*x)*log((b*x + a)/(d*x + c))^2 + (B^2*b^2*c^2 - 2*B^2*a*b*c*d + B^2*a^2 
*d^2)*n + 2*(A*B*b^2*c*d - A*B*a*b*d^2)*x + (2*A*B*b^2*d^2*x^2 + 2*A*B*a*b 
*c*d + (B^2*b^2*c^2 - B^2*a^2*d^2)*n + 2*(A*B*b^2*c*d + A*B*a*b*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*(A^2*a* 
b*c*d + (B^2*b^2*c^2 + B^2*a^2*d^2)*n^2 + (2*B^2*b^2*d^2*n^2 + A^2*b^2*d^2 
)*x^2 + (A*B*b^2*c^2 - A*B*a^2*d^2)*n + (A^2*b^2*c*d + A^2*a*b*d^2 + 2*(B^ 
2*b^2*c*d + B^2*a*b*d^2)*n^2 + 2*(A*B*b^2*c*d - A*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...
 

Sympy [F(-1)]

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

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(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. 2006 vs. \(2 (390) = 780\).

Time = 0.16 (sec) , antiderivative size = 2006, normalized size of antiderivative = 5.12 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-B^2*((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)^2 - 2*A*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) - 2/3*((3*b^2 
*c^2 - 3*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)^3 + 3*(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(b*x + a)*l 
og(d*x + c)^2 - (b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)*log(d*x + 
c)^3 + 6*(b^2*c*d - a*b*d^2)*x + 6*(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b 
*d^2)*x)*log(b*x + a) - 3*(2*b^2*d^2*x^2 + 2*a*b*c*d + (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*c*d + a*b*d^2)*x)*log 
(d*x + c))*n^2/(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*...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 2172, normalized size of antiderivative = 5.54 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*log(a + b*x)*a**4*b*c*d**2*n - 6*log(a + b*x)*a**4*b*d**3*n*x - 6*lo 
g(a + b*x)*a**3*b**2*c**2*d*n + 12*log(a + b*x)*a**3*b**2*c*d**2*n**2 - 12 
*log(a + b*x)*a**3*b**2*c*d**2*n*x + 12*log(a + b*x)*a**3*b**2*d**3*n**2*x 
 - 6*log(a + b*x)*a**3*b**2*d**3*n*x**2 - 12*log(a + b*x)*a**2*b**3*c**2*d 
*n**2 - 6*log(a + b*x)*a**2*b**3*c**2*d*n*x - 12*log(a + b*x)*a**2*b**3*c* 
d**2*n**3 - 6*log(a + b*x)*a**2*b**3*c*d**2*n*x**2 - 12*log(a + b*x)*a**2* 
b**3*d**3*n**3*x + 12*log(a + b*x)*a**2*b**3*d**3*n**2*x**2 - 12*log(a + b 
*x)*a*b**4*c**2*d*n**3 - 12*log(a + b*x)*a*b**4*c**2*d*n**2*x - 24*log(a + 
 b*x)*a*b**4*c*d**2*n**3*x - 12*log(a + b*x)*a*b**4*c*d**2*n**2*x**2 - 12* 
log(a + b*x)*a*b**4*d**3*n**3*x**2 - 12*log(a + b*x)*b**5*c**2*d*n**3*x - 
12*log(a + b*x)*b**5*c*d**2*n**3*x**2 + 6*log(c + d*x)*a**4*b*c*d**2*n + 6 
*log(c + d*x)*a**4*b*d**3*n*x + 6*log(c + d*x)*a**3*b**2*c**2*d*n - 12*log 
(c + d*x)*a**3*b**2*c*d**2*n**2 + 12*log(c + d*x)*a**3*b**2*c*d**2*n*x - 1 
2*log(c + d*x)*a**3*b**2*d**3*n**2*x + 6*log(c + d*x)*a**3*b**2*d**3*n*x** 
2 + 12*log(c + d*x)*a**2*b**3*c**2*d*n**2 + 6*log(c + d*x)*a**2*b**3*c**2* 
d*n*x + 12*log(c + d*x)*a**2*b**3*c*d**2*n**3 + 6*log(c + d*x)*a**2*b**3*c 
*d**2*n*x**2 + 12*log(c + d*x)*a**2*b**3*d**3*n**3*x - 12*log(c + d*x)*a** 
2*b**3*d**3*n**2*x**2 + 12*log(c + d*x)*a*b**4*c**2*d*n**3 + 12*log(c + d* 
x)*a*b**4*c**2*d*n**2*x + 24*log(c + d*x)*a*b**4*c*d**2*n**3*x + 12*log(c 
+ d*x)*a*b**4*c*d**2*n**2*x**2 + 12*log(c + d*x)*a*b**4*d**3*n**3*x**2 ...