\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^4 (c i+d i x)} \, dx\) [193]

Optimal result

Integrand size = 45, antiderivative size = 543 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {6 b B^2 d^2 n^2 (c+d x)}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 B^2 d n^2 (c+d x)^2}{4 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {2 b^3 B^2 n^2 (c+d x)^3}{27 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {6 b B d^2 n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 B d n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {2 b^3 B n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {3 b d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^4 g^4 i n} \] Output:

-6*b*B^2*d^2*n^2*(d*x+c)/(-a*d+b*c)^4/g^4/i/(b*x+a)+3/4*b^2*B^2*d*n^2*(d*x 
+c)^2/(-a*d+b*c)^4/g^4/i/(b*x+a)^2-2/27*b^3*B^2*n^2*(d*x+c)^3/(-a*d+b*c)^4 
/g^4/i/(b*x+a)^3-6*b*B*d^2*n*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d 
+b*c)^4/g^4/i/(b*x+a)+3/2*b^2*B*d*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^ 
n))/(-a*d+b*c)^4/g^4/i/(b*x+a)^2-2/9*b^3*B*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/ 
(d*x+c))^n))/(-a*d+b*c)^4/g^4/i/(b*x+a)^3-3*b*d^2*(d*x+c)*(A+B*ln(e*((b*x+ 
a)/(d*x+c))^n))^2/(-a*d+b*c)^4/g^4/i/(b*x+a)+3/2*b^2*d*(d*x+c)^2*(A+B*ln(e 
*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^4/g^4/i/(b*x+a)^2-1/3*b^3*(d*x+c)^3*(A 
+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^4/g^4/i/(b*x+a)^3-1/3*d^3*(A+B* 
ln(e*((b*x+a)/(d*x+c))^n))^3/B/(-a*d+b*c)^4/g^4/i/n
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1295\) vs. \(2(543)=1086\).

Time = 1.73 (sec) , antiderivative size = 1295, normalized size of antiderivative = 2.38 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/108*(36*B^2*d^3*n^2*(a + b*x)^3*Log[(a + b*x)/(c + d*x)]^3 + 18*B*n*Log 
[(a + b*x)/(c + d*x)]^2*(6*a^3*A*d^3 + 2*b^3*B*c^3*n - 9*a*b^2*B*c^2*d*n + 
 18*a^2*b*B*c*d^2*n + 18*a^2*A*b*d^3*x - 3*b^3*B*c^2*d*n*x + 18*a*b^2*B*c* 
d^2*n*x + 18*a^2*b*B*d^3*n*x + 18*a*A*b^2*d^3*x^2 + 6*b^3*B*c*d^2*n*x^2 + 
27*a*b^2*B*d^3*n*x^2 + 6*A*b^3*d^3*x^3 + 11*b^3*B*d^3*n*x^3 + 6*B*d^3*(a + 
 b*x)^3*Log[e*((a + b*x)/(c + d*x))^n] - 6*B*d^3*n*(a + b*x)^3*Log[(a + b* 
x)/(c + d*x)]) - 3*d*(b*c - a*d)^2*(a + b*x)*(18*A^2 + 30*A*B*n + 19*B^2*n 
^2 + 18*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(6*A + 5*B*n)*Log[(a 
+ b*x)/(c + d*x)] + 18*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 6*B*Log[e*((a 
+ b*x)/(c + d*x))^n]*(6*A + 5*B*n - 6*B*n*Log[(a + b*x)/(c + d*x)])) + 6*d 
^2*(b*c - a*d)*(a + b*x)^2*(18*A^2 + 66*A*B*n + 85*B^2*n^2 + 18*B^2*Log[e* 
((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(6*A + 11*B*n)*Log[(a + b*x)/(c + d*x)] 
 + 18*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 6*B*Log[e*((a + b*x)/(c + d*x)) 
^n]*(6*A + 11*B*n - 6*B*n*Log[(a + b*x)/(c + d*x)])) + 6*d^3*(a + b*x)^3*L 
og[a + b*x]*(18*A^2 + 66*A*B*n + 85*B^2*n^2 + 18*B^2*Log[e*((a + b*x)/(c + 
 d*x))^n]^2 - 6*B*n*(6*A + 11*B*n)*Log[(a + b*x)/(c + d*x)] + 18*B^2*n^2*L 
og[(a + b*x)/(c + d*x)]^2 + 6*B*Log[e*((a + b*x)/(c + d*x))^n]*(6*A + 11*B 
*n - 6*B*n*Log[(a + b*x)/(c + d*x)])) + 4*(b*c - a*d)^3*(9*A^2 + 6*A*B*n + 
 2*B^2*n^2 + 9*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(3*A + B*n)*Lo 
g[(a + b*x)/(c + d*x)] + 9*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 6*B*Log...
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.74, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 (a+b x)^3}-\frac {2 b^3 B n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 (a+b x)^3}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 (a+b x)^2}+\frac {3 b^2 B d n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\frac {d^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B n}-\frac {3 b d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{a+b x}-\frac {6 b B d^2 n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {2 b^3 B^2 n^2 (c+d x)^3}{27 (a+b x)^3}+\frac {3 b^2 B^2 d n^2 (c+d x)^2}{4 (a+b x)^2}-\frac {6 b B^2 d^2 n^2 (c+d x)}{a+b x}}{g^4 i (b c-a d)^4}\)

Input:

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

Output:

((-6*b*B^2*d^2*n^2*(c + d*x))/(a + b*x) + (3*b^2*B^2*d*n^2*(c + d*x)^2)/(4 
*(a + b*x)^2) - (2*b^3*B^2*n^2*(c + d*x)^3)/(27*(a + b*x)^3) - (6*b*B*d^2* 
n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) + (3*b^2*B*d 
*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + b*x)^2) - ( 
2*b^3*B*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(a + b*x) 
^3) - (3*b*d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b* 
x) + (3*b^2*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(a 
+ b*x)^2) - (b^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3* 
(a + b*x)^3) - (d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*n))/((b 
*c - a*d)^4*g^4*i)
 

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. \(2215\) vs. \(2(529)=1058\).

Time = 28.33 (sec) , antiderivative size = 2216, normalized size of antiderivative = 4.08

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(2216\)

Input:

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i),x,method=_ 
RETURNVERBOSE)
 

Output:

-1/108*(-54*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^5*b^3*c^4*d*n+36*B^2*ln(e* 
((b*x+a)/(d*x+c))^n)^3*a^8*c^2*d^3+108*A^2*ln(e*((b*x+a)/(d*x+c))^n)*a^8*c 
^2*d^3+36*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)^3*a^5*b^3*c^2*d^3+575*B^2*x^3* 
a^5*b^3*c^2*d^3*n^3-648*B^2*x^3*a^4*b^4*c^3*d^2*n^3+81*B^2*x^3*a^3*b^5*c^4 
*d*n^3-24*A*B*x^3*a^2*b^6*c^5*n^2+108*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^3* 
a^6*b^2*c^2*d^3+1215*B^2*x^2*a^6*b^2*c^2*d^3*n^3-1434*B^2*x^2*a^5*b^3*c^3* 
d^2*n^3+243*B^2*x^2*a^4*b^4*c^4*d*n^3+108*A^2*x^3*ln(e*((b*x+a)/(d*x+c))^n 
)*a^5*b^3*c^2*d^3+198*A^2*x^3*a^5*b^3*c^2*d^3*n-324*A^2*x^3*a^4*b^4*c^3*d^ 
2*n+162*A^2*x^3*a^3*b^5*c^4*d*n-72*A*B*x^2*a^3*b^5*c^5*n^2+108*B^2*x*ln(e* 
((b*x+a)/(d*x+c))^n)^3*a^7*b*c^2*d^3+648*B^2*x*a^7*b*c^2*d^3*n^3-810*B^2*x 
*a^6*b^2*c^3*d^2*n^3+186*B^2*x*a^5*b^3*c^4*d*n^3+324*A^2*x^2*ln(e*((b*x+a) 
/(d*x+c))^n)*a^6*b^2*c^2*d^3+486*A^2*x^2*a^6*b^2*c^2*d^3*n-864*A^2*x^2*a^5 
*b^3*c^3*d^2*n+486*A^2*x^2*a^4*b^4*c^4*d*n-72*A*B*x*a^4*b^4*c^5*n^2+324*B^ 
2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^7*b*c^3*d^2*n-162*B^2*ln(e*((b*x+a)/(d*x+c 
))^n)^2*a^6*b^2*c^4*d*n+648*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b*c^3*d^2*n^ 
2-162*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^2*c^4*d*n^2+324*A^2*x*ln(e*((b*x 
+a)/(d*x+c))^n)*a^7*b*c^2*d^3+324*A^2*x*a^7*b*c^2*d^3*n-648*A^2*x*a^6*b^2* 
c^3*d^2*n+432*A^2*x*a^5*b^3*c^4*d*n+72*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b 
^3*c^5*n-90*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b^3*c^4*d*n^2+324*A*B*x*ln 
(e*((b*x+a)/(d*x+c))^n)^2*a^7*b*c^2*d^3+648*A*B*x*a^7*b*c^2*d^3*n^2-972...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1910 vs. \(2 (529) = 1058\).

Time = 0.12 (sec) , antiderivative size = 1910, normalized size of antiderivative = 3.52 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i),x, 
algorithm="fricas")
 

Output:

-1/108*(36*A^2*b^3*c^3 - 162*A^2*a*b^2*c^2*d + 324*A^2*a^2*b*c*d^2 - 198*A 
^2*a^3*d^3 + 36*(B^2*b^3*d^3*n^2*x^3 + 3*B^2*a*b^2*d^3*n^2*x^2 + 3*B^2*a^2 
*b*d^3*n^2*x + B^2*a^3*d^3*n^2)*log((b*x + a)/(d*x + c))^3 + (8*B^2*b^3*c^ 
3 - 81*B^2*a*b^2*c^2*d + 648*B^2*a^2*b*c*d^2 - 575*B^2*a^3*d^3)*n^2 + 6*(1 
8*A^2*b^3*c*d^2 - 18*A^2*a*b^2*d^3 + 85*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*n^ 
2 + 66*(A*B*b^3*c*d^2 - A*B*a*b^2*d^3)*n)*x^2 + 18*(2*B^2*b^3*c^3 - 9*B^2* 
a*b^2*c^2*d + 18*B^2*a^2*b*c*d^2 - 11*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 - 6*B^2*a*b^2*c*d^2 + 5*B^2*a^2*b*d^3)* 
x + 6*(B^2*b^3*d^3*x^3 + 3*B^2*a*b^2*d^3*x^2 + 3*B^2*a^2*b*d^3*x + B^2*a^3 
*d^3)*log((b*x + a)/(d*x + c)))*log(e)^2 + 18*(6*A*B*a^3*d^3*n + (11*B^2*b 
^3*d^3*n^2 + 6*A*B*b^3*d^3*n)*x^3 + (2*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d + 1 
8*B^2*a^2*b*c*d^2)*n^2 + 3*(6*A*B*a*b^2*d^3*n + (2*B^2*b^3*c*d^2 + 9*B^2*a 
*b^2*d^3)*n^2)*x^2 + 3*(6*A*B*a^2*b*d^3*n - (B^2*b^3*c^2*d - 6*B^2*a*b^2*c 
*d^2 - 6*B^2*a^2*b*d^3)*n^2)*x)*log((b*x + a)/(d*x + c))^2 + 6*(4*A*B*b^3* 
c^3 - 27*A*B*a*b^2*c^2*d + 108*A*B*a^2*b*c*d^2 - 85*A*B*a^3*d^3)*n - 3*(18 
*A^2*b^3*c^2*d - 108*A^2*a*b^2*c*d^2 + 90*A^2*a^2*b*d^3 + (19*B^2*b^3*c^2* 
d - 378*B^2*a*b^2*c*d^2 + 359*B^2*a^2*b*d^3)*n^2 + 6*(5*A*B*b^3*c^2*d - 54 
*A*B*a*b^2*c*d^2 + 49*A*B*a^2*b*d^3)*n)*x + 6*(12*A*B*b^3*c^3 - 54*A*B*a*b 
^2*c^2*d + 108*A*B*a^2*b*c*d^2 - 66*A*B*a^3*d^3 + 6*(6*A*B*b^3*c*d^2 - 6*A 
*B*a*b^2*d^3 + 11*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*n)*x^2 + 18*(B^2*b^3*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**4/(d*i*x+c*i),x 
)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3445 vs. \(2 (529) = 1058\).

Time = 0.31 (sec) , antiderivative size = 3445, normalized size of antiderivative = 6.34 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i),x, 
algorithm="maxima")
 

Output:

-1/6*B^2*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d 
 - 5*a*b*d^2)*x)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3 
)*g^4*i*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d 
^3)*g^4*i*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b 
*d^3)*g^4*i*x + (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)* 
g^4*i) + 6*d^3*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^4*i) - 6*d^3*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^4*i))*log(e*(b*x/( 
d*x + c) + a/(d*x + c))^n)^2 - 1/3*A*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b 
*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^3 - 3*a*b^5*c^2*d + 
 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4*i*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d 
 + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*g^4*i*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c 
^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*g^4*i*x + (a^3*b^3*c^3 - 3*a^4*b^2*c^2 
*d + 3*a^5*b*c*d^2 - a^6*d^3)*g^4*i) + 6*d^3*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^4*i) - 6*d^3*lo 
g(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^4*i))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - 1/108*((8*b^3*c 
^3 - 81*a*b^2*c^2*d + 648*a^2*b*c*d^2 - 575*a^3*d^3 + 36*(b^3*d^3*x^3 + 3* 
a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^3 - 36*(b^3*d^3*x^3 
+ 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c)^3 + 510*(b^3*...
 

Giac [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Timed out} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i),x, 
algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 31.81 (sec) , antiderivative size = 1921, normalized size of antiderivative = 3.54 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Too large to display} \] Input:

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

Output:

((198*A^2*a^2*d^2 + 36*A^2*b^2*c^2 + 575*B^2*a^2*d^2*n^2 + 8*B^2*b^2*c^2*n 
^2 - 126*A^2*a*b*c*d + 510*A*B*a^2*d^2*n + 24*A*B*b^2*c^2*n - 73*B^2*a*b*c 
*d*n^2 - 138*A*B*a*b*c*d*n)/(6*(a*d - b*c)) + (x^2*(18*A^2*b^2*d^2 + 85*B^ 
2*b^2*d^2*n^2 + 66*A*B*b^2*d^2*n))/(a*d - b*c) + (x*(90*A^2*a*b*d^2 - 18*A 
^2*b^2*c*d + 359*B^2*a*b*d^2*n^2 - 19*B^2*b^2*c*d*n^2 + 294*A*B*a*b*d^2*n 
- 30*A*B*b^2*c*d*n))/(2*(a*d - b*c)))/(x*(54*a^4*b*d^2*g^4*i + 54*a^2*b^3* 
c^2*g^4*i - 108*a^3*b^2*c*d*g^4*i) + x^2*(54*a*b^4*c^2*g^4*i + 54*a^3*b^2* 
d^2*g^4*i - 108*a^2*b^3*c*d*g^4*i) + x^3*(18*b^5*c^2*g^4*i + 18*a^2*b^3*d^ 
2*g^4*i - 36*a*b^4*c*d*g^4*i) + 18*a^5*d^2*g^4*i + 18*a^3*b^2*c^2*g^4*i - 
36*a^4*b*c*d*g^4*i) - log(e*((a + b*x)/(c + d*x))^n)^2*((d^3*(11*B^2*n + 6 
*A*B))/(6*g^4*i*n*(a*d - b*c)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b 
*c*d^2)) - (B^2*d^3*(x*(b*((g^4*i*n*(a*d - b*c)*(3*a*d - b*c))/(6*d^2) + ( 
a*g^4*i*n*(a*d - b*c))/(3*d)) + (2*a*b*g^4*i*n*(a*d - b*c))/(3*d) + (b*g^4 
*i*n*(a*d - b*c)*(3*a*d - b*c))/(3*d^2)) + a*((g^4*i*n*(a*d - b*c)*(3*a*d 
- b*c))/(6*d^2) + (a*g^4*i*n*(a*d - b*c))/(3*d)) + (g^4*i*n*(a*d - b*c)*(3 
*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/(3*d^3) + (b^2*g^4*i*n*x^2*(a*d - b*c))/d 
))/(g^4*i*n*(a*d - b*c)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 
)*(a^3*g^4*i + b^3*g^4*i*x^3 + 3*a^2*b*g^4*i*x + 3*a*b^2*g^4*i*x^2))) - lo 
g(e*((a + b*x)/(c + d*x))^n)*((6*B^2*a*d*n - 3*B^2*b*c*n + 3*B^2*b*d*n*x)/ 
(x*(9*a^4*b*d^2*g^4*i + 9*a^2*b^3*c^2*g^4*i - 18*a^3*b^2*c*d*g^4*i) + x...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 2656, normalized size of antiderivative = 4.89 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*(108*log(a + b*x)*a**6*d**3*n + 324*log(a + b*x)*a**5*b*d**3*n**2 + 324 
*log(a + b*x)*a**5*b*d**3*n*x + 72*log(a + b*x)*a**4*b**2*c*d**2*n**2 + 37 
8*log(a + b*x)*a**4*b**2*d**3*n**3 + 972*log(a + b*x)*a**4*b**2*d**3*n**2* 
x + 324*log(a + b*x)*a**4*b**2*d**3*n*x**2 + 132*log(a + b*x)*a**3*b**3*c* 
d**2*n**3 + 216*log(a + b*x)*a**3*b**3*c*d**2*n**2*x + 1134*log(a + b*x)*a 
**3*b**3*d**3*n**3*x + 972*log(a + b*x)*a**3*b**3*d**3*n**2*x**2 + 108*log 
(a + b*x)*a**3*b**3*d**3*n*x**3 + 396*log(a + b*x)*a**2*b**4*c*d**2*n**3*x 
 + 216*log(a + b*x)*a**2*b**4*c*d**2*n**2*x**2 + 1134*log(a + b*x)*a**2*b* 
*4*d**3*n**3*x**2 + 324*log(a + b*x)*a**2*b**4*d**3*n**2*x**3 + 396*log(a 
+ b*x)*a*b**5*c*d**2*n**3*x**2 + 72*log(a + b*x)*a*b**5*c*d**2*n**2*x**3 + 
 378*log(a + b*x)*a*b**5*d**3*n**3*x**3 + 132*log(a + b*x)*b**6*c*d**2*n** 
3*x**3 - 108*log(c + d*x)*a**6*d**3*n - 324*log(c + d*x)*a**5*b*d**3*n**2 
- 324*log(c + d*x)*a**5*b*d**3*n*x - 72*log(c + d*x)*a**4*b**2*c*d**2*n**2 
 - 378*log(c + d*x)*a**4*b**2*d**3*n**3 - 972*log(c + d*x)*a**4*b**2*d**3* 
n**2*x - 324*log(c + d*x)*a**4*b**2*d**3*n*x**2 - 132*log(c + d*x)*a**3*b* 
*3*c*d**2*n**3 - 216*log(c + d*x)*a**3*b**3*c*d**2*n**2*x - 1134*log(c + d 
*x)*a**3*b**3*d**3*n**3*x - 972*log(c + d*x)*a**3*b**3*d**3*n**2*x**2 - 10 
8*log(c + d*x)*a**3*b**3*d**3*n*x**3 - 396*log(c + d*x)*a**2*b**4*c*d**2*n 
**3*x - 216*log(c + d*x)*a**2*b**4*c*d**2*n**2*x**2 - 1134*log(c + d*x)*a* 
*2*b**4*d**3*n**3*x**2 - 324*log(c + d*x)*a**2*b**4*d**3*n**2*x**3 - 39...