\[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx \]
Optimal antiderivative \[ -\frac {B \,d^{2} n \left (b x +a \right )}{\left (-a d +b c \right )^{3} g^{2} i^{2} \left (d x +c \right )}-\frac {b^{2} B n \left (d x +c \right )}{\left (-a d +b c \right )^{3} g^{2} i^{2} \left (b x +a \right )}+\frac {d^{2} \left (b x +a \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (-a d +b c \right )^{3} g^{2} i^{2} \left (d x +c \right )}-\frac {b^{2} \left (d x +c \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (-a d +b c \right )^{3} g^{2} i^{2} \left (b x +a \right )}-\frac {2 b d \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right ) \ln \left (\frac {b x +a}{d x +c}\right )}{\left (-a d +b c \right )^{3} g^{2} i^{2}}+\frac {b B d n \ln \left (\frac {b x +a}{d x +c}\right )^{2}}{\left (-a d +b c \right )^{3} g^{2} i^{2}} \]
command
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^2/(d*i*x+c*i)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ {\left (\frac {{\left (d x + c\right )} B n \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )} g^{2}} + \frac {{\left (B n + A + B\right )} {\left (d x + c\right )}}{{\left (b x + a\right )} g^{2}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________