\[ \int \frac {x^2 (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 a b}{c^{2} \sqrt {c \,x^{2}}}-\frac {a^{2}}{2 c^{2} x \sqrt {c \,x^{2}}}+\frac {b^{2} x \ln \left (x \right )}{c^{2} \sqrt {c \,x^{2}}} \]
command
integrate(x^2*(b*x+a)^2/(c*x^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {b^{2} \log \left ({\left | x \right |}\right )}{c^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} - \frac {4 \, a b \sqrt {c} x + a^{2} \sqrt {c}}{2 \, c^{3} x^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________