\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx \]
Optimal antiderivative \[ \frac {5 d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{12 e}+\frac {\left (-e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4 e}+\frac {5 d^{4} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e}+\frac {5 d^{2} x \sqrt {-e^{2} x^{2}+d^{2}}}{8} \]
command
integrate((-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
\[ -\frac {{\left (240 \, d^{5} \arctan \left (\sqrt {\frac {2 \, d}{x e + d} - 1}\right ) e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + \frac {{\left (15 \, d^{5} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 73 \, d^{5} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 55 \, d^{5} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 15 \, d^{5} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}^{4}}{d^{4}}\right )} e^{\left (-6\right )}}{192 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________