\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c i+d i x)^2} \, dx \]
Optimal antiderivative \[ \frac {3 B \left (-a d +b c \right )^{2} g^{3} n \left (b x +a \right )}{d^{3} i^{2} \left (d x +c \right )}-\frac {\left (-a d +b c \right )^{2} g^{3} \left (5 B n +6 A \right ) \left (b x +a \right )}{2 d^{3} i^{2} \left (d x +c \right )}-\frac {3 B \left (-a d +b c \right )^{2} g^{3} \left (b x +a \right ) \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{d^{3} i^{2} \left (d x +c \right )}+\frac {g^{3} \left (b x +a \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{2 d \,i^{2} \left (d x +c \right )}-\frac {\left (-a d +b c \right ) g^{3} \left (b x +a \right )^{2} \left (3 A +B n +3 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{2 d^{2} i^{2} \left (d x +c \right )}-\frac {b \left (-a d +b c \right )^{2} g^{3} \left (6 A +5 B n +6 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right ) \ln \left (\frac {-a d +b c}{b \left (d x +c \right )}\right )}{2 d^{4} i^{2}}-\frac {3 b B \left (-a d +b c \right )^{2} g^{3} n \polylog \left (2, \frac {d \left (b x +a \right )}{b \left (d x +c \right )}\right )}{d^{4} i^{2}} \]
command
integrate((b*g*x+a*g)^3*(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
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________