\[ \int x^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {9 d^{9} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{512 e^{4}}+\frac {9 d^{7} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{640 e^{4}}-\frac {20 d^{4} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{143 e^{3}}-\frac {9 d^{3} x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{40 e^{2}}-\frac {45 d^{2} x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{143 e}-\frac {d \,x^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4}-\frac {e \,x^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{13}-\frac {d^{5} \left (27027 e x +12800 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{320320 e^{5}}+\frac {27 d^{13} \arctan \! \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{1024 e^{5}}+\frac {27 d^{11} x \sqrt {-e^{2} x^{2}+d^{2}}}{1024 e^{4}} \]
command
integrate(x**4*(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
\[ \text {output too large to display} \]