\[ \int \frac {A+B x^2}{x \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {A}{3 b \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {\left (-4 A c +3 b B \right ) \left (2 c \,x^{2}+b \right )}{3 b^{3} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate((B*x^2+A)/x/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (B b c - A c^{2}\right )} x}{\sqrt {c x^{2} + b} b^{3} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b \sqrt {c} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A c^{\frac {3}{2}} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{2} \sqrt {c} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b c^{\frac {3}{2}} + 3 \, B b^{3} \sqrt {c} - 5 \, A b^{2} c^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{3} b^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x}\,{d x} \]________________________________________________________________________________________