\[ \int \frac {A+B x^2}{x^3 \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {A}{5 b \,x^{4} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {6 A c -5 b B}{15 b^{2} x^{2} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {4 c \left (-6 A c +5 b B \right ) \left (2 c \,x^{2}+b \right )}{15 b^{4} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate((B*x^2+A)/x^3/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (B b c^{2} - A c^{3}\right )} x}{\sqrt {c x^{2} + b} b^{4} \mathrm {sgn}\left (x\right )} - \frac {2 \, {\left (15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} B b c^{\frac {3}{2}} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} A c^{\frac {5}{2}} - 90 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} B b^{2} c^{\frac {3}{2}} + 90 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} A b c^{\frac {5}{2}} + 160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b^{3} c^{\frac {3}{2}} - 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A b^{2} c^{\frac {5}{2}} - 110 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{4} c^{\frac {3}{2}} + 150 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b^{3} c^{\frac {5}{2}} + 25 \, B b^{5} c^{\frac {3}{2}} - 33 \, A b^{4} c^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{5} b^{3} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]________________________________________________________________________________________