\[ \int \frac {(a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c i+d i x} \, dx \]
Optimal antiderivative \[ \frac {g \left (b x +a \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{d i}+\frac {\left (-a d +b c \right ) g \left (A +B n +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right ) \ln \left (\frac {-a d +b c}{b \left (d x +c \right )}\right )}{d^{2} i}+\frac {B \left (-a d +b c \right ) g n \polylog \left (2, \frac {d \left (b x +a \right )}{b \left (d x +c \right )}\right )}{d^{2} i} \]
command
integrate((b*g*x+a*g)*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(d*i*x+c*i),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________