\[ \int \frac {x^m \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (A b -B a \right ) x^{1+m}}{3 a b \left (b \,x^{3}+a \right )}+\frac {\left (A b \left (2-m \right )+a B \left (1+m \right )\right ) x^{1+m} \hypergeom \left (\left [1, \frac {1}{3}+\frac {m}{3}\right ], \left [\frac {4}{3}+\frac {m}{3}\right ], -\frac {b \,x^{3}}{a}\right )}{3 a^{2} b \left (1+m \right )} \]
command
integrate(x**m*(B*x**3+A)/(b*x**3+a)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________