\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx \]
Optimal antiderivative \[ \frac {B \,d^{3} \left (b x +a \right )^{2}}{4 \left (-a d +b c \right )^{4} g^{2} i^{3} \left (d x +c \right )^{2}}-\frac {3 b B \,d^{2} \left (b x +a \right )}{\left (-a d +b c \right )^{4} g^{2} i^{3} \left (d x +c \right )}-\frac {b^{3} B \left (d x +c \right )}{\left (-a d +b c \right )^{4} g^{2} i^{3} \left (b x +a \right )}+\frac {3 b^{2} B d \ln \! \left (\frac {b x +a}{d x +c}\right )^{2}}{2 \left (-a d +b c \right )^{4} g^{2} i^{3}}-\frac {d^{3} \left (b x +a \right )^{2} \left (A +B \ln \! \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{2 \left (-a d +b c \right )^{4} g^{2} i^{3} \left (d x +c \right )^{2}}+\frac {3 b \,d^{2} \left (b x +a \right ) \left (A +B \ln \! \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (-a d +b c \right )^{4} g^{2} i^{3} \left (d x +c \right )}-\frac {b^{3} \left (d x +c \right ) \left (A +B \ln \! \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (-a d +b c \right )^{4} g^{2} i^{3} \left (b x +a \right )}-\frac {3 b^{2} d \ln \! \left (\frac {b x +a}{d x +c}\right ) \left (A +B \ln \! \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}{\left (-a d +b c \right )^{4} g^{2} i^{3}} \]
command
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**2/(d*i*x+c*i)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {output too large to display} \]