13.23 Problem number 456

\[ \int \frac {(f+g x) \left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3} \, dx \]

Optimal antiderivative \[ -\frac {3 b \left (-d g +e f \right ) n x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{2 d^{2} e \left (e x +d \right )}+\frac {f^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{2 d^{2} \left (-d g +e f \right )}-\frac {\left (g x +f \right )^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{2 \left (-d g +e f \right ) \left (e x +d \right )^{2}}+\frac {3 b^{2} \left (-d g +e f \right ) n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (1+\frac {e x}{d}\right )}{d^{2} e^{2}}-\frac {3 b \left (d g +e f \right ) n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (1+\frac {e x}{d}\right )}{2 d^{2} e^{2}}+\frac {3 b^{3} \left (-d g +e f \right ) n^{3} \polylog \left (2, -\frac {e x}{d}\right )}{d^{2} e^{2}}-\frac {3 b^{2} \left (d g +e f \right ) n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {e x}{d}\right )}{d^{2} e^{2}}+\frac {3 b^{3} \left (d g +e f \right ) n^{3} \polylog \left (3, -\frac {e x}{d}\right )}{d^{2} e^{2}} \]

command

int((g*x+f)*(a+b*ln(c*x^n))^3/(e*x+d)^3,x,method=_RETURNVERBOSE)

Maple 2022.1 output

method result size
risch \(\text {Expression too large to display}\) \(11535\)

Maple 2021.1 output

\[ \int \frac {\left (g x +f \right ) \left (b \ln \left (c \,x^{n}\right )+a \right )^{3}}{\left (e x +d \right )^{3}}\, dx \]________________________________________________________________________________________