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