\[ \int \frac {1}{x (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {-\frac {2 e x}{5}+\frac {2 d}{5}}{d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {-8 e x +5 d}{15 d^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{d^{5}}+\frac {-16 e x +15 d}{15 d^{5} \sqrt {-e^{2} x^{2}+d^{2}}} \]
command
integrate(1/x/(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
\[ -\frac {1}{120} \, {\left (\frac {120 \, e^{\left (-4\right )} \log \left (\sqrt {\frac {2 \, d}{x e + d} - 1} + 1\right )}{d^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {120 \, e^{\left (-4\right )} \log \left ({\left | \sqrt {\frac {2 \, d}{x e + d} - 1} - 1 \right |}\right )}{d^{5} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {15 \, e^{\left (-4\right )}}{d^{5} \sqrt {\frac {2 \, d}{x e + d} - 1} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (3 \, d^{20} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{16} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 25 \, d^{20} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{16} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 165 \, d^{20} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{16} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4}\right )} e^{\left (-20\right )}}{d^{25} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{5}}\right )} e^{4} - \frac {{\left (15 \, \log \left (2\right ) - 30 \, \log \left (i + 1\right ) + 32 i\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{30 \, d^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________