\[ \int \frac {1}{x^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 e \left (-e x +d \right )}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {e \left (-13 e x +10 d \right )}{15 d^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 e \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{d^{6}}-\frac {e \left (-41 e x +30 d \right )}{15 d^{6} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d^{6} x} \]
command
integrate(1/x^2/(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
\[ \frac {1}{120} \, {\left ({\left (\frac {240 \, e^{\left (-5\right )} \log \left (\sqrt {\frac {2 \, d}{x e + d} - 1} + 1\right )}{d^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {240 \, e^{\left (-5\right )} \log \left ({\left | \sqrt {\frac {2 \, d}{x e + d} - 1} - 1 \right |}\right )}{d^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} + \frac {30 \, {\left (\frac {17 \, d}{x e + d} - 9\right )} e^{\left (-5\right )}}{{\left ({\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} - \sqrt {\frac {2 \, d}{x e + d} - 1}\right )} d^{6} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (3 \, d^{24} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{20} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 35 \, d^{24} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{20} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 345 \, d^{24} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{20} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4}\right )} e^{\left (-25\right )}}{d^{30} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{5}}\right )} e^{7} + \frac {8 \, {\left (15 \, e^{2} \log \left (2\right ) - 30 \, e^{2} \log \left (i + 1\right ) + 56 i \, e^{2}\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{6}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________