\[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {x \left (b \,x^{2}+a \right )}{4 a \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {3}{2}}}+\frac {3 x \left (b \,x^{2}+a \right )^{2}}{8 a^{2} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {3}{2}}}+\frac {3 \left (b \,x^{2}+a \right )^{3} \arctan \left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{8 a^{\frac {5}{2}} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {3}{2}} \sqrt {b}} \]
command
integrate(1/(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
\[ \frac {3 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {3 \, b x^{3} + 5 \, a x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________