\[ \int \frac {A+B x^2}{x^5 \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {A}{7 b \,x^{6} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {8 A c -7 b B}{35 b^{2} x^{4} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {2 c \left (-8 A c +7 b B \right )}{35 b^{3} x^{2} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {8 c^{2} \left (-8 A c +7 b B \right ) \left (2 c \,x^{2}+b \right )}{35 b^{5} \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate((B*x^2+A)/x^5/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (B b c^{3} - A c^{4}\right )} x}{\sqrt {c x^{2} + b} b^{5} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} B b c^{\frac {5}{2}} - 35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} A c^{\frac {7}{2}} - 280 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} B b^{2} c^{\frac {5}{2}} + 280 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} A b c^{\frac {7}{2}} + 1015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} B b^{3} c^{\frac {5}{2}} - 1015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} A b^{2} c^{\frac {7}{2}} - 1680 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} B b^{4} c^{\frac {5}{2}} + 2240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} A b^{3} c^{\frac {7}{2}} + 1337 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b^{5} c^{\frac {5}{2}} - 1673 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A b^{4} c^{\frac {7}{2}} - 504 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{6} c^{\frac {5}{2}} + 616 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b^{5} c^{\frac {7}{2}} + 77 \, B b^{7} c^{\frac {5}{2}} - 93 \, A b^{6} c^{\frac {7}{2}}\right )}}{35 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{7} b^{4} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{5}}\,{d x} \]________________________________________________________________________________________