\[ \int \frac {x^2 (d+e x)}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {x^{2} \left (e x +d \right )}{3 d e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2}{3 e^{3} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(x^2*(e*x+d)/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {{\left (x^{2} {\left (\frac {x}{d} + 3 \, e^{\left (-1\right )}\right )} - 2 \, d^{2} e^{\left (-3\right )}\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{3 \, {\left (x^{2} e^{2} - d^{2}\right )}^{2}} \]