\[ \int \left (a+b x^3\right )^m \left (c+d x^3\right )^2 \, dx \]
Optimal antiderivative \[ -\frac {d \left (4 a d -b c \left (10+3 m \right )\right ) x \left (b \,x^{3}+a \right )^{1+m}}{b^{2} \left (9 m^{2}+33 m +28\right )}+\frac {d x \left (b \,x^{3}+a \right )^{1+m} \left (d \,x^{3}+c \right )}{b \left (7+3 m \right )}+\frac {\left (4 a^{2} d^{2}-2 a b c d \left (7+3 m \right )+b^{2} c^{2} \left (9 m^{2}+33 m +28\right )\right ) x \left (b \,x^{3}+a \right )^{m} \hypergeom \left (\left [\frac {1}{3}, -m \right ], \left [\frac {4}{3}\right ], -\frac {b \,x^{3}}{a}\right ) \left (1+\frac {b \,x^{3}}{a}\right )^{-m}}{b^{2} \left (9 m^{2}+33 m +28\right )} \]
command
integrate((b*x**3+a)**m*(d*x**3+c)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {a^{m} c^{2} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, - m \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {2 a^{m} c d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, - m \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {a^{m} d^{2} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{3}, - m \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________