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