\[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx \]
Optimal antiderivative \[ -\frac {11 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{35 e^{2}}+\frac {d \,x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e}-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{7}-\frac {d^{3} \left (-105 e x +88 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{420 e^{4}}-\frac {d^{7} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e^{4}}-\frac {d^{5} x \sqrt {-e^{2} x^{2}+d^{2}}}{8 e^{3}} \]
command
integrate(x^3*(-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
\[ \frac {{\left (13440 \, d^{8} \arctan \left (\sqrt {\frac {2 \, d}{x e + d} - 1}\right ) e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + \frac {{\left (105 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {13}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 3780 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {11}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 189 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 4992 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 1981 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 700 \, d^{8} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 105 \, d^{8} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{8} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}^{7}}{d^{7}}\right )} e^{\left (-12\right )}}{53760 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________