\[ \int \frac {a+b x}{x^2 \left (c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {a}{6 c^{2} x^{5} \sqrt {c \,x^{2}}}-\frac {b}{5 c^{2} x^{4} \sqrt {c \,x^{2}}} \]
command
integrate((b*x+a)/x^2/(c*x^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {6 \, b x + 5 \, a}{30 \, c^{\frac {5}{2}} x^{6} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________