\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx\) [207]

Optimal result

Integrand size = 45, antiderivative size = 562 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx=-\frac {B^2 d^3 n^2 (a+b x)^2}{4 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {6 A b B d^2 n (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}+\frac {6 b B^2 d^2 n^2 (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {2 b^3 B^2 n^2 (c+d x)}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {6 b B^2 d^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^4 g^2 i^3 (c+d x)}+\frac {B d^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {2 b^3 B n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {d^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{B (b c-a d)^4 g^2 i^3 n} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1334\) vs. \(2(562)=1124\).

Time = 1.63 (sec) , antiderivative size = 1334, normalized size of antiderivative = 2.37 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/4*(4*b^2*B^2*d*n^2*(a + b*x)*(c + d*x)^2*Log[(a + b*x)/(c + d*x)]^3 + 2 
*B*n*Log[(a + b*x)/(c + d*x)]^2*(6*a*A*b^2*c^2*d + 2*b^3*B*c^3*n - 6*a^2*b 
*B*c*d^2*n + a^3*B*d^3*n + 6*A*b^3*c^2*d*x + 12*a*A*b^2*c*d^2*x + 6*b^3*B* 
c^2*d*n*x - 12*a*b^2*B*c*d^2*n*x - 3*a^2*b*B*d^3*n*x + 12*A*b^3*c*d^2*x^2 
+ 6*a*A*b^2*d^3*x^2 - 9*a*b^2*B*d^3*n*x^2 + 6*A*b^3*d^3*x^3 - 3*b^3*B*d^3* 
n*x^3 + 6*b^2*B*d*(a + b*x)*(c + d*x)^2*Log[e*((a + b*x)/(c + d*x))^n] - 6 
*b^2*B*d*n*(a + b*x)*(c + d*x)^2*Log[(a + b*x)/(c + d*x)]) + 4*b^2*(b*c - 
a*d)*(c + d*x)^2*(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)])) + 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*x)]) + d*(b*c - a*d)*(a + b*x)*(2*A - B*n + 2*B*Log 
[e*((a + b*x)/(c + d*x))^n] - 2*B*n*Log[(a + b*x)/(c + d*x)]) + 4*b^2*(c + 
 d*x)^2*(A + B*n + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c 
 + d*x)])) + d*(b*c - a*d)^2*(a + b*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^2*d*(a + b*x)* 
(c + d*x)^2*Log[a + b*x]*(2*A^2 - 2*A*B*n + 5*B^2*n^2 + 2*B^2*Log[e*((a...
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 403, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

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 
)^3),x]
 

Output:

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

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. \(1999\) vs. \(2(556)=1112\).

Time = 30.48 (sec) , antiderivative size = 2000, normalized size of antiderivative = 3.56

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(2000\)

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)^3,x,method 
=_RETURNVERBOSE)
 

Output:

-1/4*(B^2*a^3*b^3*d^7*n^3+8*B^2*b^6*c^3*d^4*n^3+2*A^2*a^3*b^3*d^7*n+4*A^2* 
b^6*c^3*d^4*n+4*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)^3*b^6*d^7+12*A^2*x^3*ln( 
e*((b*x+a)/(d*x+c))^n)*b^6*d^7-6*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^4 
*d^7*n+12*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*b^6*c^2*d^5*n+18*B^2*x*ln(e*(( 
b*x+a)/(d*x+c))^n)*a^2*b^4*d^7*n^2+24*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*b^6* 
c^2*d^5*n^2-18*B^2*x*a*b^5*c*d^6*n^3+12*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2* 
b^6*c^2*d^5+18*A*B*x*a^2*b^4*d^7*n^2-6*A*B*x*b^6*c^2*d^5*n^2-12*B^2*ln(e*( 
(b*x+a)/(d*x+c))^n)^2*a^2*b^4*c*d^6*n+24*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2 
*b^4*c*d^6*n^2+24*A^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c*d^6-12*A^2*x*a*b 
^5*c*d^6*n+12*A*B*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b^5*c^2*d^5+4*A*B*ln(e*((b 
*x+a)/(d*x+c))^n)*a^3*b^3*d^7*n+8*A*B*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^3*d^ 
4*n-12*A*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^6*d^7*n-18*B^2*x^2*ln(e*((b*x+a 
)/(d*x+c))^n)^2*a*b^5*d^7*n+42*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*d^7 
*n^2+48*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c*d^6*n^2+12*A*B*x^2*ln(e*(( 
b*x+a)/(d*x+c))^n)^2*a*b^5*d^7+24*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^6* 
c*d^6+12*A*B*x^2*a*b^5*d^7*n^2-12*A*B*x^2*b^6*c*d^6*n^2+8*B^2*x*ln(e*((b*x 
+a)/(d*x+c))^n)^3*a*b^5*c*d^6-36*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*d 
^7*n-24*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b^5*c*d^6*n+48*B^2*x*ln(e*((b* 
x+a)/(d*x+c))^n)*a*b^5*c*d^6*n^2+24*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b^ 
5*c*d^6-12*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*d^7*n+24*A*B*x*ln(e*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2057 vs. \(2 (556) = 1112\).

Time = 0.13 (sec) , antiderivative size = 2057, normalized size of antiderivative = 3.66 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/4*(4*A^2*b^3*c^3 + 6*A^2*a*b^2*c^2*d - 12*A^2*a^2*b*c*d^2 + 2*A^2*a^3*d 
^3 + 4*(B^2*b^3*d^3*n^2*x^3 + B^2*a*b^2*c^2*d*n^2 + (2*B^2*b^3*c*d^2 + B^2 
*a*b^2*d^3)*n^2*x^2 + (B^2*b^3*c^2*d + 2*B^2*a*b^2*c*d^2)*n^2*x)*log((b*x 
+ a)/(d*x + c))^3 + (8*B^2*b^3*c^3 + 15*B^2*a*b^2*c^2*d - 24*B^2*a^2*b*c*d 
^2 + 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* 
(2*B^2*b^3*c^3 + 3*B^2*a*b^2*c^2*d - 6*B^2*a^2*b*c*d^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*(3*B^2*b^3*c^2*d - 2*B^2*a*b^2*c*d^ 
2 - B^2*a^2*b*d^3)*x + 6*(B^2*b^3*d^3*x^3 + B^2*a*b^2*c^2*d + (2*B^2*b^3*c 
*d^2 + B^2*a*b^2*d^3)*x^2 + (B^2*b^3*c^2*d + 2*B^2*a*b^2*c*d^2)*x)*log((b* 
x + a)/(d*x + c)))*log(e)^2 + 2*(6*A*B*a*b^2*c^2*d*n - 3*(B^2*b^3*d^3*n^2 
- 2*A*B*b^3*d^3*n)*x^3 + (2*B^2*b^3*c^3 - 6*B^2*a^2*b*c*d^2 + B^2*a^3*d^3) 
*n^2 - 3*(3*B^2*a*b^2*d^3*n^2 - 2*(2*A*B*b^3*c*d^2 + A*B*a*b^2*d^3)*n)*x^2 
 + 3*((2*B^2*b^3*c^2*d - 4*B^2*a*b^2*c*d^2 - B^2*a^2*b*d^3)*n^2 + 2*(A*B*b 
^3*c^2*d + 2*A*B*a*b^2*c*d^2)*n)*x)*log((b*x + a)/(d*x + c))^2 + 2*(4*A*B* 
b^3*c^3 - 15*A*B*a*b^2*c^2*d + 12*A*B*a^2*b*c*d^2 - A*B*a^3*d^3)*n + 3*(6* 
A^2*b^3*c^2*d - 4*A^2*a*b^2*c*d^2 - 2*A^2*a^2*b*d^3 + (13*B^2*b^3*c^2*d - 
6*B^2*a*b^2*c*d^2 - 7*B^2*a^2*b*d^3)*n^2 - 2*(A*B*b^3*c^2*d + 2*A*B*a*b^2* 
c*d^2 - 3*A*B*a^2*b*d^3)*n)*x + 2*(4*A*B*b^3*c^3 + 6*A*B*a*b^2*c^2*d - 12* 
A*B*a^2*b*c*d^2 + 2*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. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**2/(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. 4199 vs. \(2 (556) = 1112\).

Time = 0.34 (sec) , antiderivative size = 4199, normalized size of antiderivative = 7.47 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\int { \frac {{\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{2} {\left (d i x + c i\right )}^{3}} \,d x } \] 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)^3,x 
, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 31.76 (sec) , antiderivative size = 1785, normalized size of antiderivative = 3.18 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 3638, normalized size of antiderivative = 6.47 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

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