\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x^3}\right )^p\right ) \, dx \]
Optimal antiderivative \[ -\frac {3 e \,f^{2} p \left (f x \right )^{-2+m} \hypergeom \left (\left [1, \frac {2}{3}-\frac {m}{3}\right ], \left [\frac {5}{3}-\frac {m}{3}\right ], -\frac {e}{d \,x^{3}}\right )}{d \left (-m^{2}+m +2\right )}+\frac {\left (f x \right )^{1+m} \ln \left (c \left (d +\frac {e}{x^{3}}\right )^{p}\right )}{f \left (1+m \right )} \]
command
integrate((f*x)**m*ln(c*(d+e/x**3)**p),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ 3 e p \left (\begin {cases} 0^{m} \operatorname {RootSum} {\left (27 t^{3} d e^{2} - 1, \left ( t \mapsto t \log {\left (3 t e + x \right )} \right )\right )} & \text {for}\: f = 0 \wedge m \neq -1 \\0^{m} \operatorname {RootSum} {\left (27 t^{3} d e^{2} - 1, \left ( t \mapsto t \log {\left (3 t e + x \right )} \right )\right )} & \text {for}\: f = 0 \\\frac {f f^{m} m x^{m} \Phi \left (\frac {e e^{i \pi }}{d x^{3}}, 1, \frac {2}{3} - \frac {m}{3}\right ) \Gamma \left (\frac {2}{3} - \frac {m}{3}\right )}{9 d f m x^{2} \Gamma \left (\frac {5}{3} - \frac {m}{3}\right ) + 9 d f x^{2} \Gamma \left (\frac {5}{3} - \frac {m}{3}\right )} - \frac {2 f f^{m} x^{m} \Phi \left (\frac {e e^{i \pi }}{d x^{3}}, 1, \frac {2}{3} - \frac {m}{3}\right ) \Gamma \left (\frac {2}{3} - \frac {m}{3}\right )}{9 d f m x^{2} \Gamma \left (\frac {5}{3} - \frac {m}{3}\right ) + 9 d f x^{2} \Gamma \left (\frac {5}{3} - \frac {m}{3}\right )} & \text {for}\: m > -\infty \wedge m < \infty \wedge m \neq -1 \\\frac {\begin {cases} - \frac {1}{9 d x^{3}} & \text {for}\: e = 0 \\\frac {\begin {cases} \frac {\operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x^{3}}\right )}{3} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} + \frac {\operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x^{3}}\right )}{3} & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} + \frac {\operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x^{3}}\right )}{3} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + \frac {\operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x^{3}}\right )}{3} & \text {otherwise} \end {cases}}{3 e} & \text {otherwise} \end {cases}}{f} - \frac {\left (\begin {cases} \frac {1}{3 d x^{3}} & \text {for}\: e = 0 \\\frac {\log {\left (d + \frac {e}{x^{3}} \right )}}{3 e} & \text {otherwise} \end {cases}\right ) \log {\left (f x \right )}}{f} & \text {otherwise} \end {cases}\right ) + \left (\begin {cases} 0^{m} x & \text {for}\: f = 0 \\\frac {\begin {cases} \frac {\left (f x\right )^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (f x \right )} & \text {otherwise} \end {cases}}{f} & \text {otherwise} \end {cases}\right ) \log {\left (c \left (d + \frac {e}{x^{3}}\right )^{p} \right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________