\[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^{3/2} x^8} \, dx \]
Optimal antiderivative \[ -\frac {15 a^{2} \arctanh \left (\frac {\sqrt {b}}{x \sqrt {a +\frac {b}{x^{2}}}}\right )}{8 b^{\frac {7}{2}}}+\frac {1}{b \,x^{5} \sqrt {a +\frac {b}{x^{2}}}}-\frac {5 \sqrt {a +\frac {b}{x^{2}}}}{4 b^{2} x^{3}}+\frac {15 a \sqrt {a +\frac {b}{x^{2}}}}{8 b^{3} x} \]
command
integrate(1/(a+b/x^2)^(3/2)/x^8,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {15 \, a^{2} \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right )}{8 \, \sqrt {-b} b^{3} \mathrm {sgn}\left (x\right )} + \frac {a^{2}}{\sqrt {a x^{2} + b} b^{3} \mathrm {sgn}\left (x\right )} + \frac {7 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{2} - 9 \, \sqrt {a x^{2} + b} a^{2} b}{8 \, a^{2} b^{3} x^{4} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________