\[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {5}{8 a^{2} x \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {1}{4 a x \left (b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}-\frac {15 \left (b \,x^{2}+a \right )}{8 a^{3} x \sqrt {\left (b \,x^{2}+a \right )^{2}}}-\frac {15 \left (b \,x^{2}+a \right ) \arctan \left (\frac {x \sqrt {b}}{\sqrt {a}}\right ) \sqrt {b}}{8 a^{\frac {7}{2}} \sqrt {\left (b \,x^{2}+a \right )^{2}}} \]
command
integrate(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {15 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {7 \, b^{2} x^{3} + 9 \, a b x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{3} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {1}{a^{3} x \mathrm {sgn}\left (b x^{2} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________