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