\[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-4 a c +b^{2}\right )^{3}}{320 c^{4} d \left (2 c d x +b d \right )^{\frac {5}{2}}}-\frac {\left (-4 a c +b^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{64 c^{4} d^{5}}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{448 c^{4} d^{7}}-\frac {3 \left (-4 a c +b^{2}\right )^{2}}{64 c^{4} d^{3} \sqrt {2 c d x +b d}} \]
command
integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**(7/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {\left (4 a c - b^{2}\right )^{3}}{320 c^{4} d \left (b d + 2 c d x\right )^{\frac {5}{2}}} - \frac {3 \left (4 a c - b^{2}\right )^{2}}{64 c^{4} d^{3} \sqrt {b d + 2 c d x}} + \frac {\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac {3}{2}}}{192 c^{4} d^{5}} + \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}}}{448 c^{4} d^{7}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________