\[ \int \frac {A+B x^2}{x^2 \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {3 c \left (-5 A c +4 b B \right ) \arctanh \left (\frac {x \sqrt {b}}{\sqrt {c \,x^{4}+b \,x^{2}}}\right )}{8 b^{\frac {7}{2}}}-\frac {A}{4 b \,x^{3} \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {-5 A c +4 b B}{4 b^{2} x \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {3 \left (-5 A c +4 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{8 b^{3} x^{3}} \]
command
integrate((B*x^2+A)/x^2/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {3 \, {\left (4 \, B b c - 5 \, A c^{2}\right )} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{8 \, \sqrt {-b} b^{3} \mathrm {sgn}\left (x\right )} - \frac {B b c - A c^{2}}{\sqrt {c x^{2} + b} b^{3} \mathrm {sgn}\left (x\right )} - \frac {4 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B b c - 4 \, \sqrt {c x^{2} + b} B b^{2} c - 7 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} A c^{2} + 9 \, \sqrt {c x^{2} + b} A b c^{2}}{8 \, b^{3} c^{2} x^{4} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]________________________________________________________________________________________