\[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx \]
Optimal antiderivative \[ -\frac {4 d^{4} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{21 e^{4}}+\frac {5 d^{3} x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{24 e^{3}}-\frac {5 d^{2} x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{21 e^{2}}+\frac {d \,x^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4 e}-\frac {x^{6} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{9}-\frac {d^{5} \left (-315 e x +256 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2016 e^{6}}-\frac {5 d^{9} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{64 e^{6}}-\frac {5 d^{7} x \sqrt {-e^{2} x^{2}+d^{2}}}{64 e^{5}} \]
command
integrate(x^5*(-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
\[ \frac {{\left (161280 \, d^{10} \arctan \left (\sqrt {\frac {2 \, d}{x e + d} - 1}\right ) e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + \frac {{\left (315 \, d^{10} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {17}{2}} e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 18774 \, d^{10} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {15}{2}} e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 10458 \, d^{10} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {13}{2}} e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 68958 \, d^{10} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {11}{2}} e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 8192 \, d^{10} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {9}{2}} e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 32418 \, d^{10} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 10458 \, d^{10} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 2730 \, d^{10} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 315 \, d^{10} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{10} \mathrm {sgn}\left (\frac {1}{x e + d}\right )\right )} {\left (x e + d\right )}^{9}}{d^{9}}\right )} e^{\left (-16\right )}}{1032192 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________