\[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {35 d^{10} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3072 e^{5}}+\frac {7 d^{8} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{768 e^{5}}-\frac {124 d^{5} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{1287 e^{4}}-\frac {7 d^{4} x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{48 e^{3}}-\frac {31 d^{3} x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{143 e^{2}}-\frac {7 d^{2} x^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{24 e}-\frac {3 d \,x^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{13}-\frac {e \,x^{7} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{14}-\frac {d^{6} \left (63063 e x +31744 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{1153152 e^{6}}+\frac {35 d^{14} \arctan \! \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2048 e^{6}}+\frac {35 d^{12} x \sqrt {-e^{2} x^{2}+d^{2}}}{2048 e^{5}} \]
command
integrate(x**5*(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
\[ \text {output too large to display} \]