\[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx \]
Optimal antiderivative \[ -\frac {d x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{6 e}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{15 e^{2}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 e^{2} \left (e x +d \right )^{2}}-\frac {d^{5} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{4 e^{2}}-\frac {d^{3} x \sqrt {-e^{2} x^{2}+d^{2}}}{4 e} \]
command
integrate(x*(-e^2*x^2+d^2)^(5/2)/(e*x+d)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (480 \, d^{6} \arctan \left (\sqrt {\frac {2 \, d}{x e + d} - 1}\right ) e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + \frac {{\left (15 \, d^{6} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 250 \, d^{6} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 128 \, d^{6} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 70 \, d^{6} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 15 \, d^{6} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}^{5}}{d^{5}}\right )} e^{\left (-8\right )}}{960 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________