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