\[ \int x^m (3-2 a x)^{2+n} (6+4 a x)^n \, dx \]
Optimal antiderivative \[ \frac {2^{n} 9^{1+n} x^{1+m} \hypergeom \left (\left [-n , \frac {1}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}+\frac {m}{2}\right ], \frac {4 a^{2} x^{2}}{9}\right )}{1+m}-\frac {2^{2+n} 3^{1+2 n} a \,x^{2+m} \hypergeom \left (\left [-n , 1+\frac {m}{2}\right ], \left [2+\frac {m}{2}\right ], \frac {4 a^{2} x^{2}}{9}\right )}{2+m}+\frac {2^{2+n} 9^{n} a^{2} x^{3+m} \hypergeom \left (\left [-n , \frac {3}{2}+\frac {m}{2}\right ], \left [\frac {5}{2}+\frac {m}{2}\right ], \frac {4 a^{2} x^{2}}{9}\right )}{3+m} \]
command
integrate(x**m*(-2*a*x+3)**(2+n)*(4*a*x+6)**n,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 \[ \text {Timed out} \]_____________________________________________________