\(\int (a g+b g x)^2 (c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [169]

Optimal result

Integrand size = 45, antiderivative size = 819 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=-\frac {B^2 (b c-a d)^4 g^2 i^2 n^2 x}{10 b^2 d^2}-\frac {B^2 (b c-a d)^3 g^2 i^2 n^2 (c+d x)^2}{20 b d^3}+\frac {B^2 (b c-a d)^2 g^2 i^2 n^2 (c+d x)^3}{30 d^3}-\frac {B (b c-a d)^3 g^2 i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b^3 d}-\frac {B (b c-a d)^2 g^2 i^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 b^3}-\frac {B (b c-a d)^3 g^2 i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b d^3}+\frac {4 B (b c-a d)^2 g^2 i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 d^3}-\frac {b B (b c-a d) g^2 i^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 d^3}+\frac {(b c-a d)^2 g^2 i^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{30 b^3}+\frac {(b c-a d) g^2 i^2 (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{10 b^2}+\frac {g^2 i^2 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b}+\frac {B (b c-a d)^4 g^2 i^2 n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b^3 d^2}+\frac {B (b c-a d)^5 g^2 i^2 n \left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{30 b^3 d^3}+\frac {B^2 (b c-a d)^5 g^2 i^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{30 b^3 d^3}+\frac {B^2 (b c-a d)^5 g^2 i^2 n^2 \log (c+d x)}{10 b^3 d^3}+\frac {B^2 (b c-a d)^5 g^2 i^2 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{15 b^3 d^3} \] Output:

-1/10*B^2*(-a*d+b*c)^4*g^2*i^2*n^2*x/b^2/d^2-1/20*B^2*(-a*d+b*c)^3*g^2*i^2 
*n^2*(d*x+c)^2/b/d^3+1/30*B^2*(-a*d+b*c)^2*g^2*i^2*n^2*(d*x+c)^3/d^3-1/30* 
B*(-a*d+b*c)^3*g^2*i^2*n*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^3/d-1 
/15*B*(-a*d+b*c)^2*g^2*i^2*n*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^3 
-1/5*B*(-a*d+b*c)^3*g^2*i^2*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b/ 
d^3+4/15*B*(-a*d+b*c)^2*g^2*i^2*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n) 
)/d^3-1/10*b*B*(-a*d+b*c)*g^2*i^2*n*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^ 
n))/d^3+1/30*(-a*d+b*c)^2*g^2*i^2*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n) 
)^2/b^3+1/10*(-a*d+b*c)*g^2*i^2*(b*x+a)^3*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+ 
c))^n))^2/b^2+1/5*g^2*i^2*(b*x+a)^3*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^ 
n))^2/b+1/30*B*(-a*d+b*c)^4*g^2*i^2*n*(b*x+a)*(2*A+B*n+2*B*ln(e*((b*x+a)/( 
d*x+c))^n))/b^3/d^2+1/30*B*(-a*d+b*c)^5*g^2*i^2*n*(2*A+3*B*n+2*B*ln(e*((b* 
x+a)/(d*x+c))^n))*ln((-a*d+b*c)/b/(d*x+c))/b^3/d^3+1/30*B^2*(-a*d+b*c)^5*g 
^2*i^2*n^2*ln((b*x+a)/(d*x+c))/b^3/d^3+1/10*B^2*(-a*d+b*c)^5*g^2*i^2*n^2*l 
n(d*x+c)/b^3/d^3+1/15*B^2*(-a*d+b*c)^5*g^2*i^2*n^2*polylog(2,d*(b*x+a)/b/( 
d*x+c))/b^3/d^3
 

Mathematica [A] (verified)

Time = 1.08 (sec) , antiderivative size = 1254, normalized size of antiderivative = 1.53 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(g^2*i^2*(20*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d* 
x))^n])^2 + 30*d^4*(b*c - a*d)*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d* 
x))^n])^2 + 12*d^5*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 
20*B*(b*c - a*d)^3*n*(2*A*b*d*(b*c - a*d)*x + 2*B*d*(b*c - a*d)*(a + b*x)* 
Log[e*((a + b*x)/(c + d*x))^n] - d^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/( 
c + d*x))^n]) - 2*B*(b*c - a*d)^2*n*Log[c + d*x] - 2*(b*c - a*d)^2*(A + B* 
Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + B*(b*c - a*d)*n*(b*d*x + (- 
(b*c) + a*d)*Log[c + d*x]) + B*(b*c - a*d)^2*n*((2*Log[(d*(a + b*x))/(-(b* 
c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c 
- a*d)])) - 10*B*(b*c - a*d)^2*n*(6*A*b*d*(b*c - a*d)^2*x + 6*B*d*(b*c - a 
*d)^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 3*d^2*(-(b*c) + a*d)*(a + 
 b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*d^3*(a + b*x)^3*(A + B* 
Log[e*((a + b*x)/(c + d*x))^n]) - 6*B*(b*c - a*d)^3*n*Log[c + d*x] - 6*(b* 
c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + B*(b*c - 
a*d)*n*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d* 
x]) + 3*B*(b*c - a*d)^2*n*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 3*B*(b*c 
 - a*d)^3*n*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + 
d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + B*(b*c - a*d)*n*(24*A*b 
*d*(b*c - a*d)^3*x + 24*B*d*(b*c - a*d)^3*(a + b*x)*Log[e*((a + b*x)/(c + 
d*x))^n] - 12*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c ...
 

Rubi [A] (verified)

Time = 2.23 (sec) , antiderivative size = 891, normalized size of antiderivative = 1.09, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2961, 2783, 2782, 27, 1195, 2009, 2783, 2773, 49, 2009, 2781, 2784, 2784, 2754, 2838}

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.

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

\(\Big \downarrow \) 2961

\(\displaystyle g^2 i^2 (b c-a d)^5 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}\)

\(\Big \downarrow \) 2783

\(\displaystyle g^2 i^2 (b c-a d)^5 \left (-\frac {2 B n \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {2 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 2782

\(\displaystyle g^2 i^2 (b c-a d)^5 \left (-\frac {2 B n \left (-B n \int \frac {(c+d x) \left (b^2-\frac {4 d (a+b x) b}{c+d x}+\frac {6 d^2 (a+b x)^2}{(c+d x)^2}\right )}{12 d^3 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{5 b}+\frac {2 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle g^2 i^2 (b c-a d)^5 \left (-\frac {2 B n \left (-\frac {B n \int \frac {(c+d x) \left (b^2-\frac {4 d (a+b x) b}{c+d x}+\frac {6 d^2 (a+b x)^2}{(c+d x)^2}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{12 d^3}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{5 b}+\frac {2 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 1195

\(\displaystyle g^2 i^2 (b c-a d)^5 \left (-\frac {2 B n \left (-\frac {B n \int \left (\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {5 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {3 b d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {c+d x}{b^2 (a+b x)}\right )d\frac {a+b x}{c+d x}}{12 d^3}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{5 b}+\frac {2 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle g^2 i^2 (b c-a d)^5 \left (\frac {2 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}-\frac {2 B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{12 d^3}\right )}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 2783

\(\displaystyle g^2 i^2 (b c-a d)^5 \left (\frac {2 \left (-\frac {B n \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {\int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{4 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{5 b}-\frac {2 B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{12 d^3}\right )}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 2773

\(\displaystyle g^2 i^2 (b c-a d)^5 \left (\frac {2 \left (-\frac {B n \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \int \frac {(a+b x)^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}\right )}{2 b}+\frac {\int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{4 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{5 b}-\frac {2 B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{12 d^3}\right )}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 49

\(\displaystyle g^2 i^2 (b c-a d)^5 \left (\frac {2 \left (-\frac {B n \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \int \left (\frac {b^2}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 b}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )d\frac {a+b x}{c+d x}}{3 b}\right )}{2 b}+\frac {\int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{4 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{5 b}-\frac {2 B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{12 d^3}\right )}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle g^2 i^2 (b c-a d)^5 \left (\frac {2 \left (\frac {\int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{4 b}-\frac {B n \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{3 b}\right )}{2 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{5 b}-\frac {2 B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{12 d^3}\right )}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 2781

\(\displaystyle g^2 i^2 (b c-a d)^5 \left (\frac {2 \left (\frac {\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}}{4 b}-\frac {B n \left (\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{3 b}\right )}{2 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{5 b}-\frac {2 B n \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{12 d^3}\right )}{5 b}+\frac {(a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle (b c-a d)^5 g^2 i^2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{3 b}\right )}{2 b}+\frac {\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {(a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}\right )}{3 b}}{4 b}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle (b c-a d)^5 g^2 i^2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{3 b}\right )}{2 b}+\frac {\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}}{2 d}\right )}{3 b}}{4 b}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2754

\(\displaystyle (b c-a d)^5 g^2 i^2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{3 b}\right )}{2 b}+\frac {\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {2 B n \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{2 d}\right )}{3 b}}{4 b}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle (b c-a d)^5 g^2 i^2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{5 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{3 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {5}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}\right )}{12 d^3}\right )}{5 b}+\frac {2 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^3}{4 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {b^2}{2 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^3}\right )}{3 b}\right )}{2 b}+\frac {\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{d}-\frac {2 B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{2 d}\right )}{3 b}}{4 b}\right )}{5 b}\right )\)

Input:

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

Output:

(b*c - a*d)^5*g^2*i^2*(((a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) 
^2)/(5*b*(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^5) - (2*B*n*((b^2*(A + 
B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^3*(b - (d*(a + b*x))/(c + d*x))^4) 
 - (2*b*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*d^3*(b - (d*(a + b*x))/ 
(c + d*x))^3) + (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(2*d^3*(b - (d*(a + 
 b*x))/(c + d*x))^2) - (B*n*(b/(b - (d*(a + b*x))/(c + d*x))^3 - 5/(2*(b - 
 (d*(a + b*x))/(c + d*x))^2) + 1/(b*(b - (d*(a + b*x))/(c + d*x))) + Log[( 
a + b*x)/(c + d*x)]/b^2 - Log[b - (d*(a + b*x))/(c + d*x)]/b^2))/(12*d^3)) 
)/(5*b) + (2*(((a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*b* 
(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^4) - (B*n*(((a + b*x)^3*(A + B*L 
og[e*((a + b*x)/(c + d*x))^n]))/(3*b*(c + d*x)^3*(b - (d*(a + b*x))/(c + d 
*x))^3) - (B*n*(b^2/(2*d^3*(b - (d*(a + b*x))/(c + d*x))^2) - (2*b)/(d^3*( 
b - (d*(a + b*x))/(c + d*x))) - Log[b - (d*(a + b*x))/(c + d*x)]/d^3))/(3* 
b)))/(2*b) + (((a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*b* 
(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^3) - (2*B*n*(((a + b*x)^2*(A + B 
*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d*(c + d*x)^2*(b - (d*(a + b*x))/(c + 
 d*x))^2) - (((a + b*x)*(2*A + B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n]))/ 
(d*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) - (-(((2*A + 3*B*n + 2*B*Log[e 
*((a + b*x)/(c + d*x))^n])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d) - (2*B 
*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d)/d)/(2*d)))/(3*b))/(4*b))...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

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

rule 2754

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb 
ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) 
  Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, 
b, c, d, e, n}, x] && IGtQ[p, 0]
 

rule 2773

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_)^(r_.))^(q_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + 
 b*Log[c*x^n])/(d*f*(m + 1))), x] - Simp[b*(n/(d*(m + 1)))   Int[(f*x)^m*(d 
 + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && Eq 
Q[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
 

rule 2781

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a 
+ b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Simp[b*n*(p/(d*(q + 1)))   Int[(f*x) 
^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, 
d, e, f, m, n, q}, x] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 2782

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q 
_), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x)^q, x]}, Simp[(a + b*Log[c* 
x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[ 
{a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]
 

rule 2783

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

rule 2784

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* 
(x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] 
)/(e*(q + 1))), x] - Simp[f/(e*(q + 1))   Int[(f*x)^(m - 1)*(d + e*x)^(q + 
1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, 
x] && ILtQ[q, -1] && GtQ[m, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

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 [F]

\[\int \left (b g x +a g \right )^{2} \left (d i x +c i \right )^{2} {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(A^2*b^2*d^2*g^2*i^2*x^4 + A^2*a^2*c^2*g^2*i^2 + 2*(A^2*b^2*c*d + 
A^2*a*b*d^2)*g^2*i^2*x^3 + (A^2*b^2*c^2 + 4*A^2*a*b*c*d + A^2*a^2*d^2)*g^2 
*i^2*x^2 + 2*(A^2*a*b*c^2 + A^2*a^2*c*d)*g^2*i^2*x + (B^2*b^2*d^2*g^2*i^2* 
x^4 + B^2*a^2*c^2*g^2*i^2 + 2*(B^2*b^2*c*d + B^2*a*b*d^2)*g^2*i^2*x^3 + (B 
^2*b^2*c^2 + 4*B^2*a*b*c*d + B^2*a^2*d^2)*g^2*i^2*x^2 + 2*(B^2*a*b*c^2 + B 
^2*a^2*c*d)*g^2*i^2*x)*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*(A*B*b^2*d^2*g 
^2*i^2*x^4 + A*B*a^2*c^2*g^2*i^2 + 2*(A*B*b^2*c*d + A*B*a*b*d^2)*g^2*i^2*x 
^3 + (A*B*b^2*c^2 + 4*A*B*a*b*c*d + A*B*a^2*d^2)*g^2*i^2*x^2 + 2*(A*B*a*b* 
c^2 + A*B*a^2*c*d)*g^2*i^2*x)*log(e*((b*x + a)/(d*x + c))^n), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4247 vs. \(2 (786) = 1572\).

Time = 0.64 (sec) , antiderivative size = 4247, normalized size of antiderivative = 5.19 \[ \int (a g+b g x)^2 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

2/5*A*B*b^2*d^2*g^2*i^2*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/5*A 
^2*b^2*d^2*g^2*i^2*x^5 + A*B*b^2*c*d*g^2*i^2*x^4*log(e*(b*x/(d*x + c) + a/ 
(d*x + c))^n) + A*B*a*b*d^2*g^2*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c) 
)^n) + 1/2*A^2*b^2*c*d*g^2*i^2*x^4 + 1/2*A^2*a*b*d^2*g^2*i^2*x^4 + 2/3*A*B 
*b^2*c^2*g^2*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 8/3*A*B*a*b* 
c*d*g^2*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2/3*A*B*a^2*d^2*g 
^2*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/3*A^2*b^2*c^2*g^2*i^ 
2*x^3 + 4/3*A^2*a*b*c*d*g^2*i^2*x^3 + 1/3*A^2*a^2*d^2*g^2*i^2*x^3 + 2*A*B* 
a*b*c^2*g^2*i^2*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*A*B*a^2*c*d 
*g^2*i^2*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*a*b*c^2*g^2*i^2* 
x^2 + A^2*a^2*c*d*g^2*i^2*x^2 + 1/30*A*B*b^2*d^2*g^2*i^2*n*(12*a^5*log(b*x 
 + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*( 
b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c 
^4 - a^4*d^4)*x)/(b^4*d^4)) - 1/6*A*B*b^2*c*d*g^2*i^2*n*(6*a^4*log(b*x + a 
)/b^4 - 6*c^4*log(d*x + c)/d^4 + (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c 
^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3)) - 1/6*A*B*a*b* 
d^2*g^2*i^2*n*(6*a^4*log(b*x + a)/b^4 - 6*c^4*log(d*x + c)/d^4 + (2*(b^3*c 
*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d 
^3)*x)/(b^3*d^3)) + 1/3*A*B*b^2*c^2*g^2*i^2*n*(2*a^3*log(b*x + a)/b^3 - 2* 
c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g**2*( - 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c* 
x + b*d*x**2),x)*a**5*b**2*d**6*n + 20*int((log(((a + b*x)**n*e)/(c + d*x) 
**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**4*b**3*c*d**5*n - 40*int((l 
og(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a 
**3*b**4*c**2*d**4*n + 40*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c 
+ a*d*x + b*c*x + b*d*x**2),x)*a**2*b**5*c**3*d**3*n - 20*int((log(((a + b 
*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a*b**6*c**4 
*d**2*n + 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c* 
x + b*d*x**2),x)*b**7*c**5*d*n - 4*log(c + d*x)*a**6*d**5*n + 20*log(c + d 
*x)*a**5*b*c*d**4*n - 40*log(c + d*x)*a**4*b**2*c**2*d**3*n + 40*log(c + d 
*x)*a**3*b**3*c**3*d**2*n - 20*log(c + d*x)*a**2*b**4*c**4*d*n + 4*log(c + 
 d*x)*a*b**5*c**5*n + 2*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**4*b**2*c* 
d**4 - 8*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**3*b**3*c**2*d**3 - 8*log 
(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**4*c**3*d**2 - 60*log(((a + b*x) 
**n*e)/(c + d*x)**n)**2*a**2*b**4*c**2*d**3*x - 60*log(((a + b*x)**n*e)/(c 
 + d*x)**n)**2*a**2*b**4*c*d**4*x**2 - 20*log(((a + b*x)**n*e)/(c + d*x)** 
n)**2*a**2*b**4*d**5*x**3 + 2*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**5 
*c**4*d - 60*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**5*c**2*d**3*x**2 - 
 80*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**5*c*d**4*x**3 - 30*log(((a 
+ b*x)**n*e)/(c + d*x)**n)**2*a*b**5*d**5*x**4 - 20*log(((a + b*x)**n*e...