\[ \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx \]
Optimal antiderivative \[ -\frac {8 b n x}{9 d f}+\frac {4 b n \,x^{3}}{27}+\frac {2 b n \arctan \left (x \sqrt {d}\, \sqrt {f}\right )}{9 d^{\frac {3}{2}} f^{\frac {3}{2}}}+\frac {2 x \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 d f}-\frac {2 x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )}{9}-\frac {2 \arctan \left (x \sqrt {d}\, \sqrt {f}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 d^{\frac {3}{2}} f^{\frac {3}{2}}}-\frac {b n \,x^{3} \ln \left (d f \,x^{2}+1\right )}{9}+\frac {x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d f \,x^{2}+1\right )}{3}+\frac {i b n \polylog \left (2, -i x \sqrt {d}\, \sqrt {f}\right )}{3 d^{\frac {3}{2}} f^{\frac {3}{2}}}-\frac {i b n \polylog \left (2, i x \sqrt {d}\, \sqrt {f}\right )}{3 d^{\frac {3}{2}} f^{\frac {3}{2}}} \]
command
int(x^2*(a+b*ln(c*x^n))*ln(d*(1/d+f*x^2)),x,method=_RETURNVERBOSE)
Maple 2022.1 output
\[ \text {output too large to display} \]
Maple 2021.1 output \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) x^{2} \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )\, dx \]____________________________________________________________________