\[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^6} \, dx \]
Optimal antiderivative \[ -\frac {2 B^{2} d^{2} i^{2} n^{2} \left (d x +c \right )^{3}}{27 \left (-a d +b c \right )^{3} g^{6} \left (b x +a \right )^{3}}+\frac {b \,B^{2} d \,i^{2} n^{2} \left (d x +c \right )^{4}}{16 \left (-a d +b c \right )^{3} g^{6} \left (b x +a \right )^{4}}-\frac {2 b^{2} B^{2} i^{2} n^{2} \left (d x +c \right )^{5}}{125 \left (-a d +b c \right )^{3} g^{6} \left (b x +a \right )^{5}}-\frac {2 B \,d^{2} i^{2} n \left (d x +c \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{9 \left (-a d +b c \right )^{3} g^{6} \left (b x +a \right )^{3}}+\frac {b B d \,i^{2} n \left (d x +c \right )^{4} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{4 \left (-a d +b c \right )^{3} g^{6} \left (b x +a \right )^{4}}-\frac {2 b^{2} B \,i^{2} n \left (d x +c \right )^{5} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{25 \left (-a d +b c \right )^{3} g^{6} \left (b x +a \right )^{5}}-\frac {d^{2} i^{2} \left (d x +c \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{3 \left (-a d +b c \right )^{3} g^{6} \left (b x +a \right )^{3}}+\frac {b d \,i^{2} \left (d x +c \right )^{4} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{2 \left (-a d +b c \right )^{3} g^{6} \left (b x +a \right )^{4}}-\frac {b^{2} i^{2} \left (d x +c \right )^{5} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{5 \left (-a d +b c \right )^{3} g^{6} \left (b x +a \right )^{5}} \]
command
integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^6,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{54000} \, {\left (\frac {1800 \, {\left (6 \, B^{2} b^{2} n^{2} - \frac {15 \, {\left (b x + a\right )} B^{2} b d n^{2}}{d x + c} + \frac {10 \, {\left (b x + a\right )}^{2} B^{2} d^{2} n^{2}}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{\frac {{\left (b x + a\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x + a\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x + a\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}} + \frac {60 \, {\left (72 \, B^{2} b^{2} n^{2} - \frac {225 \, {\left (b x + a\right )} B^{2} b d n^{2}}{d x + c} + \frac {200 \, {\left (b x + a\right )}^{2} B^{2} d^{2} n^{2}}{{\left (d x + c\right )}^{2}} + 360 \, A B b^{2} n + 360 \, B^{2} b^{2} n - \frac {900 \, {\left (b x + a\right )} A B b d n}{d x + c} - \frac {900 \, {\left (b x + a\right )} B^{2} b d n}{d x + c} + \frac {600 \, {\left (b x + a\right )}^{2} A B d^{2} n}{{\left (d x + c\right )}^{2}} + \frac {600 \, {\left (b x + a\right )}^{2} B^{2} 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 )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x + a\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x + a\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}} + \frac {864 \, B^{2} b^{2} n^{2} - \frac {3375 \, {\left (b x + a\right )} B^{2} b d n^{2}}{d x + c} + \frac {4000 \, {\left (b x + a\right )}^{2} B^{2} d^{2} n^{2}}{{\left (d x + c\right )}^{2}} + 4320 \, A B b^{2} n + 4320 \, B^{2} b^{2} n - \frac {13500 \, {\left (b x + a\right )} A B b d n}{d x + c} - \frac {13500 \, {\left (b x + a\right )} B^{2} b d n}{d x + c} + \frac {12000 \, {\left (b x + a\right )}^{2} A B d^{2} n}{{\left (d x + c\right )}^{2}} + \frac {12000 \, {\left (b x + a\right )}^{2} B^{2} d^{2} n}{{\left (d x + c\right )}^{2}} + 10800 \, A^{2} b^{2} + 21600 \, A B b^{2} + 10800 \, B^{2} b^{2} - \frac {27000 \, {\left (b x + a\right )} A^{2} b d}{d x + c} - \frac {54000 \, {\left (b x + a\right )} A B b d}{d x + c} - \frac {27000 \, {\left (b x + a\right )} B^{2} b d}{d x + c} + \frac {18000 \, {\left (b x + a\right )}^{2} A^{2} d^{2}}{{\left (d x + c\right )}^{2}} + \frac {36000 \, {\left (b x + a\right )}^{2} A B d^{2}}{{\left (d x + c\right )}^{2}} + \frac {18000 \, {\left (b x + a\right )}^{2} B^{2} d^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x + a\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x + a\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________