\[ \int \frac {(e x)^m \left (a+b x^2\right ) \left (A+B x^2\right )}{\left (c+d x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a d +b c \right ) \left (e x \right )^{1+m} \left (B \,x^{2}+A \right )}{4 c d e \left (d \,x^{2}+c \right )^{2}}+\frac {\left (b c \left (A d \left (1+m \right )-B c \left (3+m \right )\right )+a d \left (A d \left (3-m \right )-B \left (-c m +c \right )\right )\right ) \left (e x \right )^{1+m}}{8 c^{2} d^{2} e \left (d \,x^{2}+c \right )}+\frac {\left (a d \left (1-m \right ) \left (A d \left (3-m \right )+B c \left (1+m \right )\right )+b c \left (1+m \right ) \left (A d \left (1-m \right )+B c \left (3+m \right )\right )\right ) \left (e x \right )^{1+m} \hypergeom \left (\left [1, \frac {1}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}+\frac {m}{2}\right ], -\frac {d \,x^{2}}{c}\right )}{8 c^{3} d^{2} e \left (1+m \right )} \]
command
integrate((e*x)**m*(b*x**2+a)*(B*x**2+A)/(d*x**2+c)**3,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 \[ \text {Timed out} \]_____________________________________________________