\[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)^2} \, dx \]
Optimal antiderivative \[ \frac {B \,d^{3} n \left (b x +a \right )}{\left (-a d +b c \right )^{4} g^{3} i^{2} \left (d x +c \right )}+\frac {3 b^{2} B d n \left (d x +c \right )}{\left (-a d +b c \right )^{4} g^{3} i^{2} \left (b x +a \right )}-\frac {b^{3} B n \left (d x +c \right )^{2}}{4 \left (-a d +b c \right )^{4} g^{3} i^{2} \left (b x +a \right )^{2}}-\frac {d^{3} \left (b x +a \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (-a d +b c \right )^{4} g^{3} i^{2} \left (d x +c \right )}+\frac {3 b^{2} d \left (d x +c \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (-a d +b c \right )^{4} g^{3} i^{2} \left (b x +a \right )}-\frac {b^{3} \left (d x +c \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{2 \left (-a d +b c \right )^{4} g^{3} i^{2} \left (b x +a \right )^{2}}+\frac {3 b \,d^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right ) \ln \left (\frac {b x +a}{d x +c}\right )}{\left (-a d +b c \right )^{4} g^{3} i^{2}}-\frac {3 b B \,d^{2} n \ln \left (\frac {b x +a}{d x +c}\right )^{2}}{2 \left (-a d +b c \right )^{4} g^{3} i^{2}} \]
command
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^3/(d*i*x+c*i)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{4} \, {\left (\frac {2 \, {\left (B b n - \frac {2 \, {\left (b x + a\right )} B d n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}} + \frac {B b n - \frac {4 \, {\left (b x + a\right )} B d n}{d x + c} + 2 \, A b + 2 \, B b - \frac {4 \, {\left (b x + a\right )} A d}{d x + c} - \frac {4 \, {\left (b x + a\right )} B d}{d x + c}}{\frac {{\left (b x + a\right )}^{2} b c g^{3}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{2} a d g^{3}}{{\left (d x + c\right )}^{2}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________