32.23 Problem number 142

\[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)} \, dx \]

Optimal antiderivative \[ -\frac {3 b B \,d^{2} n \left (d x +c \right )}{\left (-a d +b c \right )^{4} g^{4} i \left (b x +a \right )}+\frac {3 b^{2} B d n \left (d x +c \right )^{2}}{4 \left (-a d +b c \right )^{4} g^{4} i \left (b x +a \right )^{2}}-\frac {b^{3} B n \left (d x +c \right )^{3}}{9 \left (-a d +b c \right )^{4} g^{4} i \left (b x +a \right )^{3}}-\frac {3 b \,d^{2} \left (d x +c \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{\left (-a d +b c \right )^{4} g^{4} i \left (b x +a \right )}+\frac {3 b^{2} d \left (d x +c \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{2 \left (-a d +b c \right )^{4} g^{4} i \left (b x +a \right )^{2}}-\frac {b^{3} \left (d x +c \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{3 \left (-a d +b c \right )^{4} g^{4} i \left (b x +a \right )^{3}}-\frac {d^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right ) \ln \left (\frac {b x +a}{d x +c}\right )}{\left (-a d +b c \right )^{4} g^{4} i}+\frac {B \,d^{3} n \ln \left (\frac {b x +a}{d x +c}\right )^{2}}{2 \left (-a d +b c \right )^{4} g^{4} i} \]

command

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4/(d*i*x+c*i),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ -\frac {1}{36} \, {\left (\frac {6 \, {\left (-2 i \, B b n - \frac {3 \, {\left (-i \, b x - i \, a\right )} B d n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {-4 i \, B b n - \frac {9 \, {\left (-i \, b x - i \, a\right )} B d n}{d x + c} - 12 i \, A b - 12 i \, B b - \frac {18 \, {\left (-i \, b x - i \, a\right )} A d}{d x + c} - \frac {18 \, {\left (-i \, b x - i \, a\right )} B d}{d x + c}}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \text {Timed out} \]________________________________________________________________________________________