\[ \int x^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {41 d^{8} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{1536 e^{3}}+\frac {41 d^{6} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{1920 e^{3}}-\frac {23 d^{3} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{99 e^{2}}-\frac {41 d^{2} x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{120 e}-\frac {3 d \,x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{11}-\frac {e \,x^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{12}-\frac {d^{4} \left (28413 e x +14720 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{221760 e^{4}}+\frac {41 d^{12} \arctan \! \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{1024 e^{4}}+\frac {41 d^{10} x \sqrt {-e^{2} x^{2}+d^{2}}}{1024 e^{3}} \]
command
integrate(x**3*(e*x+d)**3*(-e**2*x**2+d**2)**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {output too large to display} \]