\[ \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/4}} \, dx \]
Optimal antiderivative \[ \frac {x \left (b \,x^{2}+a \right )}{a \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {3}{4}}} \]
command
integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^(3/4),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {1}{a \sqrt {-\frac {b x^{2} + a}{x^{2}}}} \]