\[ \int \frac {A+B x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-3 A c +2 b B \right ) \arctanh \left (\frac {x \sqrt {b}}{\sqrt {c \,x^{4}+b \,x^{2}}}\right )}{2 b^{\frac {5}{2}}}-\frac {B}{3 c x \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {3 A c -2 b B}{3 b c x \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {\left (-3 A c +2 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{2 b^{2} c \,x^{3}} \]
command
integrate((B*x^2+A)/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (2 \, B b - 3 \, A c\right )} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b} b^{2} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (c x^{2} + b\right )} B b - 2 \, B b^{2} - 3 \, {\left (c x^{2} + b\right )} A c + 2 \, A b c}{2 \, {\left ({\left (c x^{2} + b\right )}^{\frac {3}{2}} - \sqrt {c x^{2} + b} b\right )} b^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________