\[ \int \frac {1}{x^2 \sqrt {b x^2+c x^4}} \, dx \]
Optimal antiderivative \[ \frac {c \arctanh \left (\frac {x \sqrt {b}}{\sqrt {c \,x^{4}+b \,x^{2}}}\right )}{2 b^{\frac {3}{2}}}-\frac {\sqrt {c \,x^{4}+b \,x^{2}}}{2 b \,x^{3}} \]
command
integrate(1/x^2/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {c^{2} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {\sqrt {c x^{2} + b} c}{b x^{2}}}{2 \, c \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________