\[ \int \frac {x}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {4 x}{15 d^{3} e \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {1}{5 e^{2} \left (e x +d \right )^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {2}{15 d \,e^{2} \left (e x +d \right ) \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(x/(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
\[ -\frac {1}{120} \, {\left (-\frac {32 i \, \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{3}} - \frac {15}{d^{3} \sqrt {\frac {2 \, d}{x e + d} - 1} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {3 \, d^{12} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 5 \, d^{12} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} - 15 \, d^{12} \sqrt {\frac {2 \, d}{x e + d} - 1} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4}}{d^{15} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{5}}\right )} e^{\left (-2\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________