\(\int (a g+b g x) (c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [180]

Optimal result

Integrand size = 43, antiderivative size = 786 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^4 g i^3 n^2 x}{60 b^3 d}+\frac {B^2 (b c-a d)^3 g i^3 n^2 (c+d x)^2}{30 b^2 d^2}+\frac {B^2 (b c-a d)^2 g i^3 n^2 (c+d x)^3}{30 b d^2}-\frac {B (b c-a d)^4 g i^3 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 b^4 d}-\frac {B (b c-a d)^3 g i^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 b^4}+\frac {3 B (b c-a d)^3 g i^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^2 d^2}+\frac {B (b c-a d)^2 g i^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b d^2}-\frac {B (b c-a d) g i^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{10 d^2}+\frac {(b c-a d)^3 g i^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{20 b^4}+\frac {(b c-a d)^2 g i^3 (a+b x)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{10 b^3}+\frac {3 (b c-a d) g i^3 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{20 b^2}+\frac {g i^3 (a+b x)^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 b}-\frac {B (b c-a d)^5 g i^3 n \left (A+B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{10 b^4 d^2}-\frac {B^2 (b c-a d)^5 g i^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^4 d^2}-\frac {11 B^2 (b c-a d)^5 g i^3 n^2 \log (c+d x)}{60 b^4 d^2}-\frac {B^2 (b c-a d)^5 g i^3 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{10 b^4 d^2} \] Output:

1/60*B^2*(-a*d+b*c)^4*g*i^3*n^2*x/b^3/d+1/30*B^2*(-a*d+b*c)^3*g*i^3*n^2*(d 
*x+c)^2/b^2/d^2+1/30*B^2*(-a*d+b*c)^2*g*i^3*n^2*(d*x+c)^3/b/d^2-1/10*B*(-a 
*d+b*c)^4*g*i^3*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^4/d-1/10*B*(-a 
*d+b*c)^3*g*i^3*n*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^4+3/20*B*(-a 
*d+b*c)^3*g*i^3*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2/d^2+1/30*B 
*(-a*d+b*c)^2*g*i^3*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b/d^2-1/10 
*B*(-a*d+b*c)*g*i^3*n*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^2+1/20*( 
-a*d+b*c)^3*g*i^3*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^4+1/10*(-a 
*d+b*c)^2*g*i^3*(b*x+a)^2*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^3+3/ 
20*(-a*d+b*c)*g*i^3*(b*x+a)^2*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/ 
b^2+1/5*g*i^3*(b*x+a)^2*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b-1/10 
*B*(-a*d+b*c)^5*g*i^3*n*(A+B*n+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((-a*d+b*c)/ 
b/(d*x+c))/b^4/d^2-1/12*B^2*(-a*d+b*c)^5*g*i^3*n^2*ln((b*x+a)/(d*x+c))/b^4 
/d^2-11/60*B^2*(-a*d+b*c)^5*g*i^3*n^2*ln(d*x+c)/b^4/d^2-1/10*B^2*(-a*d+b*c 
)^5*g*i^3*n^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b^4/d^2
 

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 945, normalized size of antiderivative = 1.20 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(g*i^3*(-5*(b*c - a*d)*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^ 
2 + 4*b*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (5*B*(b*c - 
 a*d)^2*n*(6*A*b*d*(b*c - a*d)^2*x - 3*B*(b*c - a*d)^2*n*(b*d*x + (b*c - a 
*d)*Log[a + b*x]) - B*(b*c - a*d)*n*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 
 + 2*(b*c - a*d)^2*Log[a + b*x]) + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[e*((a 
 + b*x)/(c + d*x))^n] + 3*b^2*(b*c - a*d)*(c + d*x)^2*(A + B*Log[e*((a + b 
*x)/(c + d*x))^n]) + 2*b^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^ 
n]) + 6*(b*c - a*d)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) 
- 6*B*(b*c - a*d)^3*n*Log[c + d*x] - 3*B*(b*c - a*d)^3*n*(Log[a + b*x]*(Lo 
g[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x) 
)/(-(b*c) + a*d)])))/(3*b^4) - (B*(b*c - a*d)*n*(24*A*b*d*(b*c - a*d)^3*x 
- 12*B*(b*c - a*d)^3*n*(b*d*x + (b*c - a*d)*Log[a + b*x]) - 4*B*(b*c - a*d 
)^2*n*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log[a + b*x 
]) - B*(b*c - a*d)*n*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + d*x)^ 
2 + 2*b^3*(c + d*x)^3 + 6*(b*c - a*d)^3*Log[a + b*x]) + 24*B*d*(b*c - a*d) 
^3*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 12*b^2*(b*c - a*d)^2*(c + d* 
x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 8*b^3*(b*c - a*d)*(c + d*x)^ 
3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 6*b^4*(c + d*x)^4*(A + B*Log[e* 
((a + b*x)/(c + d*x))^n]) + 24*(b*c - a*d)^4*Log[a + b*x]*(A + B*Log[e*((a 
 + b*x)/(c + d*x))^n]) - 24*B*(b*c - a*d)^4*n*Log[c + d*x] - 12*B*(b*c ...
 

Rubi [A] (verified)

Time = 2.31 (sec) , antiderivative size = 1002, normalized size of antiderivative = 1.27, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.442, Rules used = {2961, 2783, 2782, 27, 86, 2009, 2783, 2782, 27, 86, 2009, 2783, 2773, 49, 2009, 2781, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2783

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

\(\Big \downarrow \) 2782

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2783

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

\(\Big \downarrow \) 2782

\(\displaystyle g i^3 (b c-a d)^5 \left (\frac {3 \left (-\frac {B n \left (-B n \int -\frac {(c+d x) \left (b-\frac {3 d (a+b x)}{c+d x}\right )}{6 d^2 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{2 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{5 b}-\frac {2 B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{12 d^2}\right )}{5 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle g i^3 (b c-a d)^5 \left (\frac {3 \left (-\frac {B n \left (\frac {B n \int \frac {(c+d x) \left (b-\frac {3 d (a+b x)}{c+d x}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{6 d^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{2 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{5 b}-\frac {2 B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{12 d^2}\right )}{5 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 86

\(\displaystyle g i^3 (b c-a d)^5 \left (\frac {3 \left (-\frac {B n \left (\frac {B n \int \left (\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {2 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {c+d x}{b^2 (a+b x)}\right )d\frac {a+b x}{c+d x}}{6 d^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{2 b}+\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{5 b}-\frac {2 B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{12 d^2}\right )}{5 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle g i^3 (b c-a d)^5 \left (\frac {3 \left (\frac {\int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}-\frac {B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{5 b}-\frac {2 B n \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{12 d^2}\right )}{5 b}+\frac {(a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\)

\(\Big \downarrow \) 2783

\(\displaystyle (b c-a d)^5 g i^3 \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}{5 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \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 )^3}\right )}{12 d^2}\right )}{5 b}+\frac {3 \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}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\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 {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \int \frac {(a+b x) \left (A+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 )^3}d\frac {a+b x}{c+d x}}{3 b}+\frac {\int \frac {(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) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}}{2 b}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2773

\(\displaystyle (b c-a d)^5 g i^3 \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}{5 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \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 )^3}\right )}{12 d^2}\right )}{5 b}+\frac {3 \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}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\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 {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^2 \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 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \frac {a+b x}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 b}\right )}{3 b}+\frac {\int \frac {(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) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}}{2 b}\right )}{5 b}\right )\)

\(\Big \downarrow \) 49

\(\displaystyle (b c-a d)^5 g i^3 \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}{5 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \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 )^3}\right )}{12 d^2}\right )}{5 b}+\frac {3 \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}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\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 {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^2 \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 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \left (\frac {b}{d \left (\frac {d (a+b x)}{c+d x}-b\right )^2}+\frac {1}{d \left (\frac {d (a+b x)}{c+d x}-b\right )}\right )d\frac {a+b x}{c+d x}}{2 b}\right )}{3 b}+\frac {\int \frac {(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) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}}{2 b}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle (b c-a d)^5 g i^3 \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}{5 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \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 )^3}\right )}{12 d^2}\right )}{5 b}+\frac {3 \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}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\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 {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^2 \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 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{d^2 \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^2}\right )}{2 b}\right )}{3 b}+\frac {\int \frac {(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) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 b}}{2 b}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2781

\(\displaystyle (b c-a d)^5 g i^3 \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}{5 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \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 )^3}\right )}{12 d^2}\right )}{5 b}+\frac {3 \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}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\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 {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^2 \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 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{d^2 \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^2}\right )}{2 b}\right )}{3 b}+\frac {\frac {(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+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \frac {(a+b x) \left (A+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}}{b}}{3 b}}{2 b}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2784

\(\displaystyle (b c-a d)^5 g i^3 \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}{5 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \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 )^3}\right )}{12 d^2}\right )}{5 b}+\frac {3 \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}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\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 {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^2 \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 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{d^2 \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^2}\right )}{2 b}\right )}{3 b}+\frac {\frac {(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+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {(a+b x) \left (A+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 {A+B n+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}\right )}{b}}{3 b}}{2 b}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2754

\(\displaystyle (b c-a d)^5 g i^3 \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}{5 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \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 )^3}\right )}{12 d^2}\right )}{5 b}+\frac {3 \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}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\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 {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^2 \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 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{d^2 \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^2}\right )}{2 b}\right )}{3 b}+\frac {\frac {(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+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {(a+b x) \left (A+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 {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 (A+B n+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}\right )}{b}}{3 b}}{2 b}\right )}{5 b}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle (b c-a d)^5 g i^3 \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}{5 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \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 )^3}\right )}{12 d^2}\right )}{5 b}+\frac {3 \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}{4 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\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 {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{6 d^2}\right )}{2 b}+\frac {\frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b (c+d x)^2 \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 b (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {b}{d^2 \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^2}\right )}{2 b}\right )}{3 b}+\frac {\frac {(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+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {(a+b x) \left (A+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 (A+B n+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 {B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}\right )}{b}}{3 b}}{2 b}\right )}{5 b}\right )\)

Input:

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

Output:

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

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 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

integrate((b*g*x+a*g)*(d*i*x+c*i)**3*(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. 3724 vs. \(2 (753) = 1506\).

Time = 0.62 (sec) , antiderivative size = 3724, normalized size of antiderivative = 4.74 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g*i*(6*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 - 30*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 + 60*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**3* 
b**4*c**2*d**4*n - 60*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 + 30*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 - 6*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 + 6*log(a + b*x)*a**6*d**5*n - 30*log(a + b*x)* 
a**5*b*c*d**4*n - 5*log(a + b*x)*a**5*b*d**5*n**2 + 60*log(a + b*x)*a**4*b 
**2*c**2*d**3*n + 25*log(a + b*x)*a**4*b**2*c*d**4*n**2 - 60*log(a + b*x)* 
a**3*b**3*c**3*d**2*n - 50*log(a + b*x)*a**3*b**3*c**2*d**3*n**2 + 30*log( 
a + b*x)*a**2*b**4*c**4*d*n + 50*log(a + b*x)*a**2*b**4*c**3*d**2*n**2 - 6 
*log(a + b*x)*a*b**5*c**5*n - 25*log(a + b*x)*a*b**5*c**4*d*n**2 + 5*log(a 
 + b*x)*b**6*c**5*n**2 - 3*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**4*b**2 
*c*d**4 + 12*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**3*b**3*c**2*d**3 - 1 
8*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**4*c**3*d**2 - 3*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**3*d**2*x - 90*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a 
*b**5*c**2*d**3*x**2 - 60*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**5*...