\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx \]
Optimal antiderivative \[ -\frac {B \,d^{4} \left (b x +a \right )}{\left (-a d +b c \right )^{5} g^{4} i^{2} \left (d x +c \right )}-\frac {6 b^{2} B \,d^{2} \left (d x +c \right )}{\left (-a d +b c \right )^{5} g^{4} i^{2} \left (b x +a \right )}+\frac {b^{3} B d \left (d x +c \right )^{2}}{\left (-a d +b c \right )^{5} g^{4} i^{2} \left (b x +a \right )^{2}}-\frac {b^{4} B \left (d x +c \right )^{3}}{9 \left (-a d +b c \right )^{5} g^{4} i^{2} \left (b x +a \right )^{3}}+\frac {2 b B \,d^{3} \ln \left (\frac {b x +a}{d x +c}\right )^{2}}{\left (-a d +b c \right )^{5} g^{4} i^{2}}+\frac {d^{4} \left (b x +a \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (-a d +b c \right )^{5} g^{4} i^{2} \left (d x +c \right )}-\frac {6 b^{2} d^{2} \left (d x +c \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (-a d +b c \right )^{5} g^{4} i^{2} \left (b x +a \right )}+\frac {2 b^{3} d \left (d x +c \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (-a d +b c \right )^{5} g^{4} i^{2} \left (b x +a \right )^{2}}-\frac {b^{4} \left (d x +c \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{3 \left (-a d +b c \right )^{5} g^{4} i^{2} \left (b x +a \right )^{3}}-\frac {4 b \,d^{3} \ln \left (\frac {b x +a}{d x +c}\right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (-a d +b c \right )^{5} g^{4} i^{2}} \]
command
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (6 \, B b^{2} e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {18 \, {\left (b x e + a e\right )} B b d e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {18 \, {\left (b x e + a e\right )}^{2} B d^{2} e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + 6 \, A b^{2} e^{4} + 2 \, B b^{2} e^{4} - \frac {18 \, {\left (b x e + a e\right )} A b d e^{3}}{d x + c} - \frac {9 \, {\left (b x e + a e\right )} B b d e^{3}}{d x + c} + \frac {18 \, {\left (b x e + a e\right )}^{2} A d^{2} e^{2}}{{\left (d x + c\right )}^{2}} + \frac {18 \, {\left (b x e + a e\right )}^{2} B d^{2} e^{2}}{{\left (d x + c\right )}^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2}}{18 \, {\left (\frac {{\left (b x e + a e\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b x e + a e\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b x e + a e\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________