\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx\) [200]

Optimal result

Integrand size = 45, antiderivative size = 560 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\frac {2 A B d^3 n (a+b x)}{(b c-a d)^4 g^3 i^2 (c+d x)}-\frac {2 B^2 d^3 n^2 (a+b x)}{(b c-a d)^4 g^3 i^2 (c+d x)}+\frac {6 b^2 B^2 d n^2 (c+d x)}{(b c-a d)^4 g^3 i^2 (a+b x)}-\frac {b^3 B^2 n^2 (c+d x)^2}{4 (b c-a d)^4 g^3 i^2 (a+b x)^2}+\frac {2 B^2 d^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^4 g^3 i^2 (c+d x)}+\frac {6 b^2 B d n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^3 i^2 (a+b x)}-\frac {b^3 B n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^3 i^2 (a+b x)^2}-\frac {d^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^4 g^3 i^2 (c+d x)}+\frac {3 b^2 d (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^4 g^3 i^2 (a+b x)}-\frac {b^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^4 g^3 i^2 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{B (b c-a d)^4 g^3 i^2 n} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1340\) vs. \(2(560)=1120\).

Time = 1.66 (sec) , antiderivative size = 1340, normalized size of antiderivative = 2.39 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*b*B^2*d^2*n^2*(a + b*x)^2*(c + d*x)*Log[(a + b*x)/(c + d*x)]^3 + 2*B*n* 
Log[(a + b*x)/(c + d*x)]^2*(6*a^2*A*b*c*d^2 - b^3*B*c^3*n + 6*a*b^2*B*c^2* 
d*n - 2*a^3*B*d^3*n + 12*a*A*b^2*c*d^2*x + 6*a^2*A*b*d^3*x + 3*b^3*B*c^2*d 
*n*x + 12*a*b^2*B*c*d^2*n*x - 6*a^2*b*B*d^3*n*x + 6*A*b^3*c*d^2*x^2 + 12*a 
*A*b^2*d^3*x^2 + 9*b^3*B*c*d^2*n*x^2 + 6*A*b^3*d^3*x^3 + 3*b^3*B*d^3*n*x^3 
 + 6*b*B*d^2*(a + b*x)^2*(c + d*x)*Log[e*((a + b*x)/(c + d*x))^n] - 6*b*B* 
d^2*n*(a + b*x)^2*(c + d*x)*Log[(a + b*x)/(c + d*x)]) + 2*b*d*(b*c - a*d)* 
(a + b*x)*(c + d*x)*(4*A^2 + 10*A*B*n + 11*B^2*n^2 + 4*B^2*Log[e*((a + b*x 
)/(c + d*x))^n]^2 - 2*B*n*(4*A + 5*B*n)*Log[(a + b*x)/(c + d*x)] + 4*B^2*n 
^2*Log[(a + b*x)/(c + d*x)]^2 + 2*B*Log[e*((a + b*x)/(c + d*x))^n]*(4*A + 
5*B*n - 4*B*n*Log[(a + b*x)/(c + d*x)])) - b*(b*c - a*d)^2*(c + d*x)*(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)])) + 6*b*d^2*(a + b*x)^2*(c + d*x)*Log[a + b*x]*(2*A^2 + 2*A*B*n + 5* 
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)])) + 2*B* 
(b*c - a*d)*n*Log[(a + b*x)/(c + d*x)]*(2*b*d*(a + b*x)*(c + d*x)*(4*A + 5 
*B*n + 4*B*Log[e*((a + b*x)/(c + d*x))^n] - 4*B*n*Log[(a + b*x)/(c + d*...
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 401, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((2*A*B*d^3*n*(a + b*x))/(c + d*x) - (2*B^2*d^3*n^2*(a + b*x))/(c + d*x) + 
 (6*b^2*B^2*d*n^2*(c + d*x))/(a + b*x) - (b^3*B^2*n^2*(c + d*x)^2)/(4*(a + 
 b*x)^2) + (2*B^2*d^3*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(c + d*x 
) + (6*b^2*B*d*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b* 
x) - (b^3*B*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + 
b*x)^2) - (d^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(c + d* 
x) + (3*b^2*d*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x 
) - (b^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(a + b*x 
)^2) + (b*d^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(B*n))/((b*c - a*d 
)^4*g^3*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. \(1985\) vs. \(2(554)=1108\).

Time = 30.55 (sec) , antiderivative size = 1986, normalized size of antiderivative = 3.55

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1986\)

Input:

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

Output:

1/4*(36*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^7*c*d^5*n+48*A*B*x*ln(e*((b*x+ 
a)/(d*x+c))^n)*a*b^6*c*d^5*n+4*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)^3*b^7*d^6 
+12*A^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^7*d^6-8*B^2*a^3*b^4*d^6*n^3-B^2*b^ 
7*c^3*d^3*n^3-4*A^2*a^3*b^4*d^6*n-2*A^2*b^7*c^3*d^3*n-30*A*B*a^2*b^5*c*d^5 
*n^2+12*A*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^7*d^6*n+18*B^2*x^2*ln(e*((b*x+ 
a)/(d*x+c))^n)^2*b^7*c*d^5*n+48*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^6*d^ 
6*n^2+24*A*B*a*b^6*c^2*d^4*n^2+18*A*B*x*b^7*c^2*d^4*n^2+12*B^2*ln(e*((b*x+ 
a)/(d*x+c))^n)^2*a*b^6*c^2*d^4*n+24*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^6*c^ 
2*d^4*n^2+24*A^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^6*c*d^5+12*A^2*x*a*b^6*c* 
d^5*n+12*A*B*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^5*c*d^5-8*A*B*ln(e*((b*x+a) 
/(d*x+c))^n)*a^3*b^4*d^6*n-4*A*B*ln(e*((b*x+a)/(d*x+c))^n)*b^7*c^3*d^3*n+4 
2*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^7*c*d^5*n^2+24*A*B*x^2*ln(e*((b*x+a) 
/(d*x+c))^n)^2*a*b^6*d^6+12*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^7*c*d^5- 
12*A*B*x^2*a*b^6*d^6*n^2+12*A*B*x^2*b^7*c*d^5*n^2+8*B^2*x*ln(e*((b*x+a)/(d 
*x+c))^n)^3*a*b^6*c*d^5-12*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^5*d^6*n 
+6*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*b^7*c^2*d^4*n+24*B^2*x*ln(e*((b*x+a)/ 
(d*x+c))^n)*a^2*b^5*d^6*n^2+18*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*b^7*c^2*d^4 
*n^2+18*B^2*x*a*b^6*c*d^5*n^3+12*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^5 
*d^6-6*A*B*x*a^2*b^5*d^6*n^2-15*B^2*a^2*b^5*c*d^5*n^3+24*B^2*a*b^6*c^2*d^4 
*n^3+8*A*B*a^3*b^4*d^6*n^2-2*A*B*b^7*c^3*d^3*n^2-6*A^2*a^2*b^5*c*d^5*n+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2052 vs. \(2 (554) = 1108\).

Time = 0.13 (sec) , antiderivative size = 2052, normalized size of antiderivative = 3.66 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/4*(2*A^2*b^3*c^3 - 12*A^2*a*b^2*c^2*d + 6*A^2*a^2*b*c*d^2 + 4*A^2*a^3*d 
^3 - 4*(B^2*b^3*d^3*n^2*x^3 + B^2*a^2*b*c*d^2*n^2 + (B^2*b^3*c*d^2 + 2*B^2 
*a*b^2*d^3)*n^2*x^2 + (2*B^2*a*b^2*c*d^2 + B^2*a^2*b*d^3)*n^2*x)*log((b*x 
+ a)/(d*x + c))^3 + (B^2*b^3*c^3 - 24*B^2*a*b^2*c^2*d + 15*B^2*a^2*b*c*d^2 
 + 8*B^2*a^3*d^3)*n^2 - 6*(2*A^2*b^3*c*d^2 - 2*A^2*a*b^2*d^3 + 5*(B^2*b^3* 
c*d^2 - B^2*a*b^2*d^3)*n^2 + 2*(A*B*b^3*c*d^2 - A*B*a*b^2*d^3)*n)*x^2 + 2* 
(B^2*b^3*c^3 - 6*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2 + 2*B^2*a^3*d^3 - 6*( 
B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*x^2 - 3*(B^2*b^3*c^2*d + 2*B^2*a*b^2*c*d^2 
- 3*B^2*a^2*b*d^3)*x - 6*(B^2*b^3*d^3*x^3 + B^2*a^2*b*c*d^2 + (B^2*b^3*c*d 
^2 + 2*B^2*a*b^2*d^3)*x^2 + (2*B^2*a*b^2*c*d^2 + B^2*a^2*b*d^3)*x)*log((b* 
x + a)/(d*x + c)))*log(e)^2 - 2*(6*A*B*a^2*b*c*d^2*n + 3*(B^2*b^3*d^3*n^2 
+ 2*A*B*b^3*d^3*n)*x^3 - (B^2*b^3*c^3 - 6*B^2*a*b^2*c^2*d + 2*B^2*a^3*d^3) 
*n^2 + 3*(3*B^2*b^3*c*d^2*n^2 + 2*(A*B*b^3*c*d^2 + 2*A*B*a*b^2*d^3)*n)*x^2 
 + 3*((B^2*b^3*c^2*d + 4*B^2*a*b^2*c*d^2 - 2*B^2*a^2*b*d^3)*n^2 + 2*(2*A*B 
*a*b^2*c*d^2 + A*B*a^2*b*d^3)*n)*x)*log((b*x + a)/(d*x + c))^2 + 2*(A*B*b^ 
3*c^3 - 12*A*B*a*b^2*c^2*d + 15*A*B*a^2*b*c*d^2 - 4*A*B*a^3*d^3)*n - 3*(2* 
A^2*b^3*c^2*d + 4*A^2*a*b^2*c*d^2 - 6*A^2*a^2*b*d^3 + (7*B^2*b^3*c^2*d + 6 
*B^2*a*b^2*c*d^2 - 13*B^2*a^2*b*d^3)*n^2 + 2*(3*A*B*b^3*c^2*d - 2*A*B*a*b^ 
2*c*d^2 - A*B*a^2*b*d^3)*n)*x + 2*(2*A*B*b^3*c^3 - 12*A*B*a*b^2*c^2*d + 6* 
A*B*a^2*b*c*d^2 + 4*A*B*a^3*d^3 - 6*(2*A*B*b^3*c*d^2 - 2*A*B*a*b^2*d^3 ...
 

Sympy [F(-1)]

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

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**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. 4198 vs. \(2 (554) = 1108\).

Time = 0.34 (sec) , antiderivative size = 4198, normalized size of antiderivative = 7.50 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*B^2*((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)^2 + 
 A*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 - 24*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 8*a^3*d^3 - 4*(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...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 31.67 (sec) , antiderivative size = 1784, normalized size of antiderivative = 3.19 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(B^2*b*d^2*log(e*((a + b*x)/(c + d*x))^n)^3)/(g^3*i^2*n*(a*d - b*c)^4) - l 
og(e*((a + b*x)/(c + d*x))^n)^2*(((B^2*(2*a*d + b*c))/(2*(a^2*d^2 + b^2*c^ 
2 - 2*a*b*c*d)) + (3*B^2*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*(2*A + B*n))/(2*g^3*i^2*n* 
(a*d - b*c)^4) + (3*B^2*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^2*c*g^3*i^2 + b^2*d*g^3*i^2*x^3))) - ((4*A^2*a^2*d^2 - 
2*A^2*b^2*c^2 + 8*B^2*a^2*d^2*n^2 - B^2*b^2*c^2*n^2 + 10*A^2*a*b*c*d - 8*A 
*B*a^2*d^2*n - 2*A*B*b^2*c^2*n + 23*B^2*a*b*c*d*n^2 + 22*A*B*a*b*c*d*n)/(2 
*(a*d - b*c)) + (3*x^2*(2*A^2*b^2*d^2 + 5*B^2*b^2*d^2*n^2 + 2*A*B*b^2*d^2* 
n))/(a*d - b*c) + (3*x*(6*A^2*a*b*d^2 + 2*A^2*b^2*c*d + 13*B^2*a*b*d^2*n^2 
 + 7*B^2*b^2*c*d*n^2 + 2*A*B*a*b*d^2*n + 6*A*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^2 + 5*B^2*n^2 + 2*A*B*n)*(2*a^4*d^4*g^3*i^2 - 2*b^ 
4*c^4*g^3*i^2 + 4*a*b^3*c^3*d*g^3*i^2 - 4*a^3*b*c*d^3*g^3*i^2)*3i)/(2*g...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 3630, normalized size of antiderivative = 6.48 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 24*log(a + b*x)*a**5*b*c*d**3*n - 24*log(a + b*x)*a**5*b*d**4*n*x - 12 
*log(a + b*x)*a**4*b**2*c**2*d**2*n - 60*log(a + b*x)*a**4*b**2*c*d**3*n*x 
 - 48*log(a + b*x)*a**4*b**2*d**4*n*x**2 - 36*log(a + b*x)*a**3*b**3*c**2* 
d**2*n**2 - 24*log(a + b*x)*a**3*b**3*c**2*d**2*n*x - 48*log(a + b*x)*a**3 
*b**3*c*d**3*n**3 - 36*log(a + b*x)*a**3*b**3*c*d**3*n**2*x - 48*log(a + b 
*x)*a**3*b**3*c*d**3*n*x**2 - 48*log(a + b*x)*a**3*b**3*d**4*n**3*x - 24*l 
og(a + b*x)*a**3*b**3*d**4*n*x**3 - 42*log(a + b*x)*a**2*b**4*c**2*d**2*n* 
*3 - 72*log(a + b*x)*a**2*b**4*c**2*d**2*n**2*x - 12*log(a + b*x)*a**2*b** 
4*c**2*d**2*n*x**2 - 138*log(a + b*x)*a**2*b**4*c*d**3*n**3*x - 72*log(a + 
 b*x)*a**2*b**4*c*d**3*n**2*x**2 - 12*log(a + b*x)*a**2*b**4*c*d**3*n*x**3 
 - 96*log(a + b*x)*a**2*b**4*d**4*n**3*x**2 - 84*log(a + b*x)*a*b**5*c**2* 
d**2*n**3*x - 36*log(a + b*x)*a*b**5*c**2*d**2*n**2*x**2 - 132*log(a + b*x 
)*a*b**5*c*d**3*n**3*x**2 - 36*log(a + b*x)*a*b**5*c*d**3*n**2*x**3 - 48*l 
og(a + b*x)*a*b**5*d**4*n**3*x**3 - 42*log(a + b*x)*b**6*c**2*d**2*n**3*x* 
*2 - 42*log(a + b*x)*b**6*c*d**3*n**3*x**3 + 24*log(c + d*x)*a**5*b*c*d**3 
*n + 24*log(c + d*x)*a**5*b*d**4*n*x + 12*log(c + d*x)*a**4*b**2*c**2*d**2 
*n + 60*log(c + d*x)*a**4*b**2*c*d**3*n*x + 48*log(c + d*x)*a**4*b**2*d**4 
*n*x**2 + 36*log(c + d*x)*a**3*b**3*c**2*d**2*n**2 + 24*log(c + d*x)*a**3* 
b**3*c**2*d**2*n*x + 48*log(c + d*x)*a**3*b**3*c*d**3*n**3 + 36*log(c + d* 
x)*a**3*b**3*c*d**3*n**2*x + 48*log(c + d*x)*a**3*b**3*c*d**3*n*x**2 + ...