\[ \int \frac {(a+b x)^2}{x^3 \left (c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {a^{2}}{5 c \,x^{4} \sqrt {c \,x^{2}}}-\frac {a b}{2 c \,x^{3} \sqrt {c \,x^{2}}}-\frac {b^{2}}{3 c \,x^{2} \sqrt {c \,x^{2}}} \]
command
integrate((b*x+a)^2/x^3/(c*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {10 \, b^{2} \sqrt {c} x^{2} + 15 \, a b \sqrt {c} x + 6 \, a^{2} \sqrt {c}}{30 \, c^{2} x^{5} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (b x + a\right )}^{2}}{\left (c x^{2}\right )^{\frac {3}{2}} x^{3}}\,{d x} \]________________________________________________________________________________________