\[ \int \frac {x^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {x}{15 d^{2} e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {d}{5 e^{3} \left (e x +d \right )^{2} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {7}{15 e^{3} \left (e x +d \right ) \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(x^2/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{120} \, {\left (-\frac {8 i \, e^{\left (-2\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{2}} - \frac {15 \, e^{\left (-2\right )}}{d^{2} \sqrt {\frac {2 \, d}{x e + d} - 1} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} + \frac {{\left (3 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} - 5 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} - 15 \, d^{8} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4}\right )} e^{\left (-10\right )}}{d^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{5}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________