\[ \int x^m (3-2 a x)^{1+n} (6+4 a x)^n \, dx \]
Optimal antiderivative \[ \frac {2^{n} 3^{1+2 n} x^{1+m} \hypergeom \left (\left [-n , \frac {1}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}+\frac {m}{2}\right ], \frac {4 a^{2} x^{2}}{9}\right )}{1+m}-\frac {2^{1+n} 9^{n} a \,x^{2+m} \hypergeom \left (\left [-n , 1+\frac {m}{2}\right ], \left [2+\frac {m}{2}\right ], \frac {4 a^{2} x^{2}}{9}\right )}{2+m} \]
command
integrate(x**m*(-2*a*x+3)**(1+n)*(4*a*x+6)**n,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {9 \cdot 18^{n} 4^{- \frac {m}{2}} \cdot 9^{\frac {m}{2}} a^{- m} {G_{6, 6}^{5, 3}\left (\begin {matrix} - \frac {m}{2} - \frac {n}{2}, - \frac {m}{2} - \frac {n}{2} + \frac {1}{2}, 1 & \frac {1}{2} - \frac {m}{2}, - \frac {m}{2} - n, - \frac {m}{2} - n + \frac {1}{2} \\- \frac {m}{2} - n - \frac {1}{2}, - \frac {m}{2} - n, - \frac {m}{2} - \frac {n}{2}, - \frac {m}{2} - n + \frac {1}{2}, - \frac {m}{2} - \frac {n}{2} + \frac {1}{2} & 0 \end {matrix} \middle | {\frac {9}{4 a^{2} x^{2}}} \right )} e^{i \pi n}}{8 \pi a \Gamma \left (- n\right )} + \frac {9 \cdot 18^{n} 4^{- \frac {m}{2}} \cdot 9^{\frac {m}{2}} a^{- m} {G_{6, 6}^{5, 3}\left (\begin {matrix} - \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2}, 1 & - \frac {m}{2}, - \frac {m}{2} - n - \frac {1}{2}, - \frac {m}{2} - n \\- \frac {m}{2} - n - 1, - \frac {m}{2} - n - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, - \frac {m}{2} - n, - \frac {m}{2} - \frac {n}{2} & 0 \end {matrix} \middle | {\frac {9}{4 a^{2} x^{2}}} \right )} e^{i \pi n}}{8 \pi a \Gamma \left (- n\right )} + \frac {9 \cdot 18^{n} 4^{- \frac {m}{2}} \cdot 9^{\frac {m}{2}} a^{- m} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {m}{2} - 1, - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, - \frac {m}{2} - \frac {n}{2} - 1, - \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, 1 & \\- \frac {m}{2} - \frac {n}{2} - 1, - \frac {m}{2} - \frac {n}{2} - \frac {1}{2} & - \frac {m}{2} - 1, - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2} - n - 1, 0 \end {matrix} \middle | {\frac {9 e^{- 2 i \pi }}{4 a^{2} x^{2}}} \right )} e^{- i \pi m}}{8 \pi a \Gamma \left (- n\right )} + \frac {9 \cdot 18^{n} 4^{- \frac {m}{2}} \cdot 9^{\frac {m}{2}} a^{- m} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, \frac {1}{2} - \frac {m}{2}, - \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2}, 1 & \\- \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2} & - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, - \frac {m}{2} - n - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {9 e^{- 2 i \pi }}{4 a^{2} x^{2}}} \right )} e^{- i \pi m}}{8 \pi a \Gamma \left (- n\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________