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