\[ \int \left (f+g x^3\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx \]
Optimal antiderivative \[ -2 f^{2} p x +\frac {2 d^{3} g^{2} p x}{7 e^{3}}+\frac {d f g p \,x^{2}}{2 e}-\frac {2 d^{2} g^{2} p \,x^{3}}{21 e^{2}}-\frac {f g p \,x^{4}}{4}+\frac {2 d \,g^{2} p \,x^{5}}{35 e}-\frac {2 g^{2} p \,x^{7}}{49}-\frac {2 d^{\frac {7}{2}} g^{2} p \arctan \left (\frac {x \sqrt {e}}{\sqrt {d}}\right )}{7 e^{\frac {7}{2}}}-\frac {d^{2} f g p \ln \left (e \,x^{2}+d \right )}{2 e^{2}}+f^{2} x \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )+\frac {f g \,x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2}+\frac {g^{2} x^{7} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{7}+\frac {2 f^{2} p \arctan \left (\frac {x \sqrt {e}}{\sqrt {d}}\right ) \sqrt {d}}{\sqrt {e}} \]
command
integrate((g*x**3+f)**2*ln(c*(e*x**2+d)**p),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \left (f^{2} x + \frac {f g x^{4}}{2} + \frac {g^{2} x^{7}}{7}\right ) \log {\left (0^{p} c \right )} & \text {for}\: d = 0 \wedge e = 0 \\- 2 f^{2} p x + f^{2} x \log {\left (c \left (e x^{2}\right )^{p} \right )} - \frac {f g p x^{4}}{4} + \frac {f g x^{4} \log {\left (c \left (e x^{2}\right )^{p} \right )}}{2} - \frac {2 g^{2} p x^{7}}{49} + \frac {g^{2} x^{7} \log {\left (c \left (e x^{2}\right )^{p} \right )}}{7} & \text {for}\: d = 0 \\\left (f^{2} x + \frac {f g x^{4}}{2} + \frac {g^{2} x^{7}}{7}\right ) \log {\left (c d^{p} \right )} & \text {for}\: e = 0 \\- \frac {2 d^{4} g^{2} p \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{7 e^{4} \sqrt {- \frac {d}{e}}} + \frac {d^{4} g^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{7 e^{4} \sqrt {- \frac {d}{e}}} + \frac {2 d^{3} g^{2} p x}{7 e^{3}} - \frac {d^{2} f g \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2 e^{2}} - \frac {2 d^{2} g^{2} p x^{3}}{21 e^{2}} + \frac {2 d f^{2} p \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{e \sqrt {- \frac {d}{e}}} - \frac {d f^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{e \sqrt {- \frac {d}{e}}} + \frac {d f g p x^{2}}{2 e} + \frac {2 d g^{2} p x^{5}}{35 e} - 2 f^{2} p x + f^{2} x \log {\left (c \left (d + e x^{2}\right )^{p} \right )} - \frac {f g p x^{4}}{4} + \frac {f g x^{4} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2} - \frac {2 g^{2} p x^{7}}{49} + \frac {g^{2} x^{7} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{7} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________