\[ \int \frac {1}{x^3 (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 e^{2} \left (-e x +d \right )}{5 d^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {e^{2} \left (-6 e x +5 d \right )}{5 d^{5} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {9 e^{2} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{2 d^{7}}+\frac {2 e^{2} \left (-11 e x +10 d \right )}{5 d^{7} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{6} x^{2}}+\frac {2 e \sqrt {-e^{2} x^{2}+d^{2}}}{d^{7} x} \]
command
integrate(1/x^3/(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
\[ -\frac {1}{40} \, {\left ({\left (\frac {180 \, e^{\left (-6\right )} \log \left (\sqrt {\frac {2 \, d}{x e + d} - 1} + 1\right )}{d^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {180 \, e^{\left (-6\right )} \log \left ({\left | \sqrt {\frac {2 \, d}{x e + d} - 1} - 1 \right |}\right )}{d^{7} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {5 \, e^{\left (-6\right )}}{d^{7} \sqrt {\frac {2 \, d}{x e + d} - 1} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} + \frac {10 \, {\left (5 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {\frac {2 \, d}{x e + d} - 1}\right )} e^{\left (-6\right )}}{d^{7} {\left (\frac {d}{x e + d} - 1\right )}^{2} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {{\left (d^{28} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{24} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 15 \, d^{28} {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{24} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4} + 195 \, d^{28} \sqrt {\frac {2 \, d}{x e + d} - 1} e^{24} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{4}\right )} e^{\left (-30\right )}}{d^{35} \mathrm {sgn}\left (\frac {1}{x e + d}\right )^{5}}\right )} e^{9} + \frac {2 \, {\left (45 \, e^{3} \log \left (2\right ) - 90 \, e^{3} \log \left (i + 1\right ) + 128 i \, e^{3}\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d^{7}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________