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