\[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx \]
Optimal antiderivative \[ -\frac {\left (a B d \left (3+m \right )-b \left (2 A d +B c \left (5+m +2 p \right )\right )\right ) \left (e x \right )^{1+m} \left (b \,x^{2}+a \right )^{1+p}}{b^{2} e \left (3+m +2 p \right ) \left (5+m +2 p \right )}+\frac {d \left (e x \right )^{1+m} \left (b \,x^{2}+a \right )^{1+p} \left (B \,x^{2}+A \right )}{b e \left (5+m +2 p \right )}-\frac {\left (A b \left (3+m +2 p \right ) \left (a d \left (1+m \right )-b c \left (5+m +2 p \right )\right )-a \left (1+m \right ) \left (a B d \left (3+m \right )-b \left (2 A d +B c \left (5+m +2 p \right )\right )\right )\right ) \left (e x \right )^{1+m} \left (b \,x^{2}+a \right )^{p} \hypergeom \left (\left [-p , \frac {1}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}+\frac {m}{2}\right ], -\frac {b \,x^{2}}{a}\right ) \left (1+\frac {b \,x^{2}}{a}\right )^{-p}}{b^{2} e \left (1+m \right ) \left (3+m +2 p \right ) \left (5+m +2 p \right )} \]
command
integrate((e*x)**m*(b*x**2+a)**p*(B*x**2+A)*(d*x**2+c),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {A a^{p} c e^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {A a^{p} d e^{m} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {B a^{p} c e^{m} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {B a^{p} d e^{m} x^{5} x^{m} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {5}{2} \\ \frac {m}{2} + \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________