\[ \int \frac {1}{\left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {3 c \arctanh \left (\frac {x \sqrt {b}}{\sqrt {c \,x^{4}+b \,x^{2}}}\right )}{2 b^{\frac {5}{2}}}+\frac {1}{b x \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {3 \sqrt {c \,x^{4}+b \,x^{2}}}{2 b^{2} x^{3}} \]
command
integrate(1/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {3 \, c \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b} b^{2} \mathrm {sgn}\left (x\right )} - \frac {3 \, {\left (c x^{2} + b\right )} c - 2 \, b c}{2 \, {\left ({\left (c x^{2} + b\right )}^{\frac {3}{2}} - \sqrt {c x^{2} + b} b\right )} b^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \mathit {sage}_{0} x \]________________________________________________________________________________________