\[ \int \frac {1}{\sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\arcsinh \! \left (\frac {x \sqrt {b}}{\sqrt {a}}\right ) \sqrt {a}\, \sqrt {1+\frac {b \,x^{2}}{a}}}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {1}{4}} \sqrt {b}} \]
command
integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^(1/4),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
\[ -\frac {\arctan \left (\frac {\sqrt {-\frac {b x^{2} + a}{x^{2}}}}{\sqrt {b}}\right )}{\sqrt {b}} \]