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