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