\[ \int (f x)^m \left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right ) \, dx \]
Optimal antiderivative \[ -\frac {2 b d e n \,x^{1+r} \left (f x \right )^{m}}{\left (1+m +r \right )^{2}}-\frac {b \,e^{2} n \,x^{1+2 r} \left (f x \right )^{m}}{\left (1+m +2 r \right )^{2}}-\frac {b \,d^{2} n \left (f x \right )^{1+m}}{f \left (1+m \right )^{2}}+\frac {2 d e \,x^{1+r} \left (f x \right )^{m} \left (a +b \ln \left (c \,x^{n}\right )\right )}{1+m +r}+\frac {e^{2} x^{1+2 r} \left (f x \right )^{m} \left (a +b \ln \left (c \,x^{n}\right )\right )}{1+m +2 r}+\frac {d^{2} \left (f x \right )^{1+m} \left (a +b \ln \left (c \,x^{n}\right )\right )}{f \left (1+m \right )} \]
command
integrate((f*x)**m*(d+e*x**r)**2*(a+b*ln(c*x**n)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________