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