\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c i+d i x} \, dx \]
Optimal antiderivative \[ \frac {g^{3} \left (b x +a \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{3 d i}-\frac {\left (-a d +b c \right ) g^{3} \left (b x +a \right )^{2} \left (3 A +B +3 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{6 d^{2} i}+\frac {\left (-a d +b c \right )^{2} g^{3} \left (b x +a \right ) \left (6 A +5 B +6 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{6 d^{3} i}+\frac {\left (-a d +b c \right )^{3} g^{3} \ln \left (\frac {-a d +b c}{b \left (d x +c \right )}\right ) \left (6 A +11 B +6 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{6 d^{4} i}+\frac {B \left (-a d +b c \right )^{3} g^{3} \polylog \left (2, \frac {d \left (b x +a \right )}{b \left (d x +c \right )}\right )}{d^{4} i} \]
command
integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),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 \[ \text {Timed out} \]_______________________________________________________