\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4 x^{4}}+\frac {2 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d \,x^{3}}+\frac {5 e^{4} \arctanh \left (\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d}\right )}{8 d}-\frac {5 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{8 x^{2}} \]
command
integrate((-e^2*x^2+d^2)^(5/2)/x^5/(e*x+d)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{192} \, {\left (\frac {120 \, e^{3} \log \left (\sqrt {\frac {2 \, d}{x e + d} - 1} + 1\right ) \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d} - \frac {120 \, e^{3} \log \left ({\left | \sqrt {\frac {2 \, d}{x e + d} - 1} - 1 \right |}\right ) \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d} + \frac {4 \, {\left (15 \, e^{3} \log \left (2\right ) - 30 \, e^{3} \log \left (i + 1\right ) + 32 i \, e^{3}\right )} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d} - \frac {15 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {7}{2}} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 73 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {5}{2}} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) - 55 \, {\left (\frac {2 \, d}{x e + d} - 1\right )}^{\frac {3}{2}} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right ) + 15 \, \sqrt {\frac {2 \, d}{x e + d} - 1} e^{3} \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{d {\left (\frac {d}{x e + d} - 1\right )}^{4}}\right )} e \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________