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