\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^3 (c i+d i x)^3} \, dx \]
Optimal antiderivative \[ \frac {B^{2} d^{4} \left (b x +a \right )^{2}}{4 \left (-a d +b c \right )^{5} g^{3} i^{3} \left (d x +c \right )^{2}}+\frac {8 A b B \,d^{3} \left (b x +a \right )}{\left (-a d +b c \right )^{5} g^{3} i^{3} \left (d x +c \right )}-\frac {8 b \,B^{2} d^{3} \left (b x +a \right )}{\left (-a d +b c \right )^{5} g^{3} i^{3} \left (d x +c \right )}+\frac {8 b^{3} B^{2} d \left (d x +c \right )}{\left (-a d +b c \right )^{5} g^{3} i^{3} \left (b x +a \right )}-\frac {b^{4} B^{2} \left (d x +c \right )^{2}}{4 \left (-a d +b c \right )^{5} g^{3} i^{3} \left (b x +a \right )^{2}}+\frac {8 b \,B^{2} d^{3} \left (b x +a \right ) \ln \! \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{\left (-a d +b c \right )^{5} g^{3} i^{3} \left (d x +c \right )}-\frac {B \,d^{4} \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 )^{5} g^{3} i^{3} \left (d x +c \right )^{2}}+\frac {8 b^{3} B d \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^{3} i^{3} \left (b x +a \right )}-\frac {b^{4} B \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 )^{5} g^{3} i^{3} \left (b x +a \right )^{2}}+\frac {d^{4} \left (b x +a \right )^{2} \left (A +B \ln \! \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{2 \left (-a d +b c \right )^{5} g^{3} i^{3} \left (d x +c \right )^{2}}-\frac {4 b \,d^{3} \left (b x +a \right ) \left (A +B \ln \! \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (-a d +b c \right )^{5} g^{3} i^{3} \left (d x +c \right )}+\frac {4 b^{3} d \left (d x +c \right ) \left (A +B \ln \! \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (-a d +b c \right )^{5} g^{3} i^{3} \left (b x +a \right )}-\frac {b^{4} \left (d x +c \right )^{2} \left (A +B \ln \! \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{2 \left (-a d +b c \right )^{5} g^{3} i^{3} \left (b x +a \right )^{2}}+\frac {2 b^{2} d^{2} \left (A +B \ln \! \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{3}}{B \left (-a d +b c \right )^{5} g^{3} i^{3}} \]
command
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**3/(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
\[ \text {output too large to display} \]