\[ \int \frac {x^3}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {d^{2} \left (-e x +d \right )^{2}}{5 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d \left (-e x +d \right )}{5 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {-2 e x +5 d}{5 d \,e^{4} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(x^3/(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
\[ -\frac {1}{40} \, {\left (\frac {16 i \, e^{\left (-3\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d} - \frac {5 \, e^{\left (-3\right )}}{d \sqrt {\frac {2 \, d}{x e + d} - 1} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (d^{4} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} - 5 \, d^{4} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 15 \, d^{4} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4}\right )} e^{\left (-15\right )}}{d^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{5}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________