\[ \int \frac {a+b x}{x \left (c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {a}{5 c^{2} x^{4} \sqrt {c \,x^{2}}}-\frac {b}{4 c^{2} x^{3} \sqrt {c \,x^{2}}} \]
command
integrate((b*x+a)/x/(c*x^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {5 \, b x + 4 \, a}{20 \, c^{\frac {5}{2}} x^{5} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {b x + a}{\left (c x^{2}\right )^{\frac {5}{2}} x}\,{d x} \]________________________________________________________________________________________