\[ \int \frac {(a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c i+d i x)^2} \, dx \]
Optimal antiderivative \[ -\frac {A g \left (b x +a \right )}{d \,i^{2} \left (d x +c \right )}+\frac {B g n \left (b x +a \right )}{d \,i^{2} \left (d x +c \right )}-\frac {B g \left (b x +a \right ) \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{d \,i^{2} \left (d x +c \right )}-\frac {b g \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right ) \ln \left (\frac {-a d +b c}{b \left (d x +c \right )}\right )}{d^{2} i^{2}}-\frac {b B g n \polylog \left (2, \frac {d \left (b x +a \right )}{b \left (d x +c \right )}\right )}{d^{2} i^{2}} \]
command
integrate((b*g*x+a*g)*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(d*i*x+c*i)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{2} \, {\left (\frac {{\left (B b^{4} c^{3} g n - 3 \, B a b^{3} c^{2} d g n - \frac {2 \, {\left (b x + a\right )} B b^{3} c^{3} d g n}{d x + c} + 3 \, B a^{2} b^{2} c d^{2} g n + \frac {6 \, {\left (b x + a\right )} B a b^{2} c^{2} d^{2} g n}{d x + c} - B a^{3} b d^{3} g n - \frac {6 \, {\left (b x + a\right )} B a^{2} b c d^{3} g n}{d x + c} + \frac {2 \, {\left (b x + a\right )} B a^{3} d^{4} g n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b^{2} d^{2} - \frac {2 \, {\left (b x + a\right )} b d^{3}}{d x + c} + \frac {{\left (b x + a\right )}^{2} d^{4}}{{\left (d x + c\right )}^{2}}} + \frac {B b^{4} c^{3} g n - 3 \, B a b^{3} c^{2} d g n - \frac {{\left (b x + a\right )} B b^{3} c^{3} d g n}{d x + c} + 3 \, B a^{2} b^{2} c d^{2} g n + \frac {3 \, {\left (b x + a\right )} B a b^{2} c^{2} d^{2} g n}{d x + c} - B a^{3} b d^{3} g n - \frac {3 \, {\left (b x + a\right )} B a^{2} b c d^{3} g n}{d x + c} + \frac {{\left (b x + a\right )} B a^{3} d^{4} g n}{d x + c} + A b^{4} c^{3} g + B b^{4} c^{3} g - 3 \, A a b^{3} c^{2} d g - 3 \, B a b^{3} c^{2} d g - \frac {2 \, {\left (b x + a\right )} A b^{3} c^{3} d g}{d x + c} - \frac {2 \, {\left (b x + a\right )} B b^{3} c^{3} d g}{d x + c} + 3 \, A a^{2} b^{2} c d^{2} g + 3 \, B a^{2} b^{2} c d^{2} g + \frac {6 \, {\left (b x + a\right )} A a b^{2} c^{2} d^{2} g}{d x + c} + \frac {6 \, {\left (b x + a\right )} B a b^{2} c^{2} d^{2} g}{d x + c} - A a^{3} b d^{3} g - B a^{3} b d^{3} g - \frac {6 \, {\left (b x + a\right )} A a^{2} b c d^{3} g}{d x + c} - \frac {6 \, {\left (b x + a\right )} B a^{2} b c d^{3} g}{d x + c} + \frac {2 \, {\left (b x + a\right )} A a^{3} d^{4} g}{d x + c} + \frac {2 \, {\left (b x + a\right )} B a^{3} d^{4} g}{d x + c}}{b^{2} d^{2} - \frac {2 \, {\left (b x + a\right )} b d^{3}}{d x + c} + \frac {{\left (b x + a\right )}^{2} d^{4}}{{\left (d x + c\right )}^{2}}} + \frac {{\left (B b^{3} c^{3} g n - 3 \, B a b^{2} c^{2} d g n + 3 \, B a^{2} b c d^{2} g n - B a^{3} d^{3} g n\right )} \log \left (-b + \frac {{\left (b x + a\right )} d}{d x + c}\right )}{b d^{2}} - \frac {{\left (B b^{3} c^{3} g n - 3 \, B a b^{2} c^{2} d g n + 3 \, B a^{2} b c d^{2} g n - B a^{3} d^{3} g n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b d^{2}}\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
\[ \text {Timed out} \]________________________________________________________________________________________