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