\[ \int (f x)^{-1+m} \left (d+e x^m\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx \]
Optimal antiderivative \[ -\frac {b d n \left (f x \right )^{m}}{f \,m^{2}}-\frac {b e n \,x^{m} \left (f x \right )^{m}}{4 f \,m^{2}}+\frac {d \left (f x \right )^{m} \left (a +b \ln \left (c \,x^{n}\right )\right )}{f m}+\frac {e \,x^{m} \left (f x \right )^{m} \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 f m} \]
command
integrate((f*x)**(-1+m)*(d+e*x**m)*(a+b*ln(c*x**n)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \tilde {\infty } \left (d + e\right ) \left (a x - b n x + b x \log {\left (c x^{n} \right )}\right ) & \text {for}\: f = 0 \wedge m = 0 \\\frac {\left (d + e\right ) \left (\begin {cases} a \log {\left (x \right )} & \text {for}\: b = 0 \\- \left (- a - b \log {\left (c \right )}\right ) \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}\right )}{f} & \text {for}\: m = 0 \\0^{m - 1} \left (\frac {a d m^{2} x}{m^{2} + 2 m + 1} + \frac {2 a d m x}{m^{2} + 2 m + 1} + \frac {a d x}{m^{2} + 2 m + 1} + \frac {a e m x x^{m}}{m^{2} + 2 m + 1} + \frac {a e x x^{m}}{m^{2} + 2 m + 1} - \frac {b d m^{2} n x}{m^{2} + 2 m + 1} + \frac {b d m^{2} x \log {\left (c x^{n} \right )}}{m^{2} + 2 m + 1} - \frac {2 b d m n x}{m^{2} + 2 m + 1} + \frac {2 b d m x \log {\left (c x^{n} \right )}}{m^{2} + 2 m + 1} - \frac {b d n x}{m^{2} + 2 m + 1} + \frac {b d x \log {\left (c x^{n} \right )}}{m^{2} + 2 m + 1} + \frac {b e m x x^{m} \log {\left (c x^{n} \right )}}{m^{2} + 2 m + 1} - \frac {b e n x x^{m}}{m^{2} + 2 m + 1} + \frac {b e x x^{m} \log {\left (c x^{n} \right )}}{m^{2} + 2 m + 1}\right ) & \text {for}\: f = 0 \\\frac {a d \left (f x\right )^{m}}{f m} + \frac {a e x^{m} \left (f x\right )^{m}}{2 f m} + \frac {b d \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{f m} - \frac {b d n \left (f x\right )^{m}}{f m^{2}} + \frac {b e x^{m} \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{2 f m} - \frac {b e n x^{m} \left (f x\right )^{m}}{4 f m^{2}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________