\[ \int (g x)^m (d+e x)^3 \left (a+c x^2\right )^p \, dx \]
Optimal antiderivative \[ \frac {3 d \,e^{2} \left (g x \right )^{1+m} \left (c \,x^{2}+a \right )^{1+p}}{c g \left (3+m +2 p \right )}+\frac {e^{3} \left (g x \right )^{2+m} \left (c \,x^{2}+a \right )^{1+p}}{c \,g^{2} \left (4+m +2 p \right )}-\frac {d \left (3 a \,e^{2} \left (1+m \right )-c \,d^{2} \left (3+m +2 p \right )\right ) \left (g x \right )^{1+m} \left (c \,x^{2}+a \right )^{p} \hypergeom \left (\left [-p , \frac {1}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}+\frac {m}{2}\right ], -\frac {c \,x^{2}}{a}\right ) \left (1+\frac {c \,x^{2}}{a}\right )^{-p}}{c g \left (1+m \right ) \left (3+m +2 p \right )}-\frac {e \left (a \,e^{2} \left (2+m \right )-3 c \,d^{2} \left (4+m +2 p \right )\right ) \left (g x \right )^{2+m} \left (c \,x^{2}+a \right )^{p} \hypergeom \left (\left [-p , 1+\frac {m}{2}\right ], \left [2+\frac {m}{2}\right ], -\frac {c \,x^{2}}{a}\right ) \left (1+\frac {c \,x^{2}}{a}\right )^{-p}}{c \,g^{2} \left (2+m \right ) \left (4+m +2 p \right )} \]
command
integrate((g*x)**m*(e*x+d)**3*(c*x**2+a)**p,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {a^{p} d^{3} g^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {3 a^{p} d^{2} e g^{m} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {3 a^{p} d e^{2} g^{m} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {a^{p} e^{3} g^{m} x^{4} x^{m} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + 3\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________