\[ \int \frac {A+B x^2}{x^4 \sqrt {b x^2+c x^4}} \, dx \]
Optimal antiderivative \[ \frac {c \left (-3 A c +4 b B \right ) \arctanh \left (\frac {x \sqrt {b}}{\sqrt {c \,x^{4}+b \,x^{2}}}\right )}{8 b^{\frac {5}{2}}}-\frac {A \sqrt {c \,x^{4}+b \,x^{2}}}{4 b \,x^{5}}-\frac {\left (-3 A c +4 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{8 b^{2} x^{3}} \]
command
integrate((B*x^2+A)/x^4/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {{\left (4 \, B b c^{2} - 3 \, A c^{3}\right )} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {4 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B b c^{2} - 4 \, \sqrt {c x^{2} + b} B b^{2} c^{2} - 3 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} A c^{3} + 5 \, \sqrt {c x^{2} + b} A b c^{3}}{b^{2} c^{2} x^{4}}}{8 \, c \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________