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