\[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx \]
Optimal antiderivative \[ \frac {d^{3} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{24 e^{2}}+\frac {d \left (-5 e x +6 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{30 e^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{3}}+\frac {d^{7} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 e^{3}}+\frac {d^{5} x \sqrt {-e^{2} x^{2}+d^{2}}}{16 e^{2}} \]
command
integrate(x^2*(-e^2*x^2+d^2)^(5/2)/(e*x+d),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{16} \, d^{7} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} \mathrm {sgn}\left (d\right ) + \frac {1}{1680} \, {\left (96 \, d^{6} e^{\left (-3\right )} - {\left (105 \, d^{5} e^{\left (-2\right )} - 2 \, {\left (24 \, d^{4} e^{\left (-1\right )} + {\left (245 \, d^{3} - 4 \, {\left (48 \, d^{2} e - 5 \, {\left (6 \, x e^{3} - 7 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________