\[ \int \frac {x (a+b x)}{\left (c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {a}{3 c^{2} x^{2} \sqrt {c \,x^{2}}}-\frac {b}{2 c^{2} x \sqrt {c \,x^{2}}} \]
command
integrate(x*(b*x+a)/(c*x^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {3 \, b x + 2 \, a}{6 \, c^{\frac {5}{2}} x^{3} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (b x + a\right )} x}{\left (c x^{2}\right )^{\frac {5}{2}}}\,{d x} \]________________________________________________________________________________________