\[ \int \frac {1}{x^2 \sqrt {b x^2}} \, dx \]
Optimal antiderivative \[ -\frac {1}{2 x \sqrt {b \,x^{2}}} \]
command
integrate(1/x^2/(b*x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {1}{2 \, \sqrt {b} x^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________