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