\[ \int \frac {x^2 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx \]
Optimal antiderivative \[ \frac {2 d \,x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{5 e}-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{6}+\frac {d^{2} \left (-45 e x +32 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{120 e^{3}}+\frac {3 d^{6} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 e^{3}}+\frac {3 d^{4} x \sqrt {-e^{2} x^{2}+d^{2}}}{16 e^{2}} \]
command
integrate(x^2*(-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
\[ -\frac {{\left (2880 \, d^{7} \arctan \left (\sqrt {\frac {2 \, d}{x e + d} - 1}\right ) e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + \frac {{\left (45 \, d^{7} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {11}{2}} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 1025 \, d^{7} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 174 \, d^{7} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 594 \, d^{7} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 255 \, d^{7} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 45 \, d^{7} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}^{6}}{d^{6}}\right )} e^{\left (-10\right )}}{7680 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________