\[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 x}{5 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {1}{5 d e \left (e x +d \right )^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {1}{5 d^{2} e \left (e x +d \right ) \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{40} \, {\left ({\left (\frac {5 \, e^{\left (-3\right )}}{d^{4} \sqrt {\frac {2 \, d}{x e + d} - 1} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (d^{16} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 5 \, d^{16} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 15 \, d^{16} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{12} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4}\right )} e^{\left (-15\right )}}{d^{20} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{5}}\right )} e^{3} + \frac {16 i \, \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{4}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________