\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx \]
Optimal antiderivative \[ -\frac {3 b B \,d^{2} \left (d x +c \right )}{\left (-a d +b c \right )^{4} g^{4} i \left (b x +a \right )}+\frac {3 b^{2} B d \left (d x +c \right )^{2}}{4 \left (-a d +b c \right )^{4} g^{4} i \left (b x +a \right )^{2}}-\frac {b^{3} B \left (d x +c \right )^{3}}{9 \left (-a d +b c \right )^{4} g^{4} i \left (b x +a \right )^{3}}+\frac {B \,d^{3} \ln \left (\frac {b x +a}{d x +c}\right )^{2}}{2 \left (-a d +b c \right )^{4} g^{4} i}-\frac {3 b \,d^{2} \left (d x +c \right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (-a d +b c \right )^{4} g^{4} i \left (b x +a \right )}+\frac {3 b^{2} d \left (d x +c \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{2 \left (-a d +b c \right )^{4} g^{4} i \left (b x +a \right )^{2}}-\frac {b^{3} \left (d x +c \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{3 \left (-a d +b c \right )^{4} g^{4} i \left (b x +a \right )^{3}}-\frac {d^{3} \ln \left (\frac {b x +a}{d x +c}\right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (-a d +b c \right )^{4} g^{4} i} \]
command
integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (-12 i \, B b e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) + \frac {18 i \, {\left (b x e + a e\right )} B d e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - 12 i \, A b e^{4} - 4 i \, B b e^{4} + \frac {18 i \, {\left (b x e + a e\right )} A d e^{3}}{d x + c} + \frac {9 i \, {\left (b x e + a e\right )} B d e^{3}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2}}{36 \, {\left (\frac {{\left (b x e + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x e + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________