\[ \int \frac {1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {105}{128 a^{4} x \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {1}{8 a x \left (b \,x^{2}+a \right )^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {3}{16 a^{2} x \left (b \,x^{2}+a \right )^{2} \sqrt {\left (b \,x^{2}+a \right )^{2}}}+\frac {21}{64 a^{3} x \left (b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}-\frac {315 \left (b \,x^{2}+a \right )}{128 a^{5} x \sqrt {\left (b \,x^{2}+a \right )^{2}}}-\frac {315 \left (b \,x^{2}+a \right ) \arctan \left (\frac {x \sqrt {b}}{\sqrt {a}}\right ) \sqrt {b}}{128 a^{\frac {11}{2}} \sqrt {\left (b \,x^{2}+a \right )^{2}}} \]
command
integrate(1/x^2/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {315 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {1}{a^{5} x \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {187 \, b^{4} x^{7} + 643 \, a b^{3} x^{5} + 765 \, a^{2} b^{2} x^{3} + 325 \, a^{3} b x}{128 \, {\left (b x^{2} + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x^{2} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________