\[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx \]
Optimal antiderivative \[ -\frac {B \,d^{4} n \left (b x +a \right )}{\left (-a d +b c \right )^{5} g^{4} i^{2} \left (d x +c \right )}-\frac {6 b^{2} B \,d^{2} n \left (d x +c \right )}{\left (-a d +b c \right )^{5} g^{4} i^{2} \left (b x +a \right )}+\frac {b^{3} B d n \left (d x +c \right )^{2}}{\left (-a d +b c \right )^{5} g^{4} i^{2} \left (b x +a \right )^{2}}-\frac {b^{4} B n \left (d x +c \right )^{3}}{9 \left (-a d +b c \right )^{5} g^{4} i^{2} \left (b x +a \right )^{3}}+\frac {d^{4} \left (b x +a \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (-a d +b c \right )^{5} g^{4} i^{2} \left (d x +c \right )}-\frac {6 b^{2} d^{2} \left (d x +c \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (-a d +b c \right )^{5} g^{4} i^{2} \left (b x +a \right )}+\frac {2 b^{3} d \left (d x +c \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (-a d +b c \right )^{5} g^{4} i^{2} \left (b x +a \right )^{2}}-\frac {b^{4} \left (d x +c \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{3 \left (-a d +b c \right )^{5} g^{4} i^{2} \left (b x +a \right )^{3}}-\frac {4 b \,d^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right ) \ln \left (\frac {b x +a}{d x +c}\right )}{\left (-a d +b c \right )^{5} g^{4} i^{2}}+\frac {2 b B \,d^{3} n \ln \left (\frac {b x +a}{d x +c}\right )^{2}}{\left (-a d +b c \right )^{5} g^{4} i^{2}} \]
command
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(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
\[ \frac {1}{18} \, {\left (\frac {6 \, {\left (B b^{2} n - \frac {3 \, {\left (b x + a\right )} B b d n}{d x + c} + \frac {3 \, {\left (b x + a\right )}^{2} B d^{2} n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b x + a\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {2 \, B b^{2} n - \frac {9 \, {\left (b x + a\right )} B b d n}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} B d^{2} n}{{\left (d x + c\right )}^{2}} + 6 \, A b^{2} + 6 \, B b^{2} - \frac {18 \, {\left (b x + a\right )} A b d}{d x + c} - \frac {18 \, {\left (b x + a\right )} B b d}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} A d^{2}}{{\left (d x + c\right )}^{2}} + \frac {18 \, {\left (b x + a\right )}^{2} B d^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b x + a\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________