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