\[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {4 x}{21 d^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {1}{7 d e \left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {1}{7 d^{2} e \left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{21 d^{6} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{672} \, {\left ({\left (\frac {14 \, {\left (\frac {15 \, d}{x e + d} - 7\right )} e^{\left (-5\right )}}{d^{6} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (3 \, d^{36} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{6} + 21 \, d^{36} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{6} + 70 \, d^{36} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{6} + 210 \, d^{36} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{6}\right )} e^{\left (-35\right )}}{d^{42} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{7}}\right )} e^{5} + \frac {256 i \, \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{6}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________