\[ \int \frac {1}{x \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {1}{b \,x^{2} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {4 \sqrt {c \,x^{4}+b \,x^{2}}}{3 b^{2} x^{4}}+\frac {8 c \sqrt {c \,x^{4}+b \,x^{2}}}{3 b^{3} x^{2}} \]
command
integrate(1/x/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {c^{2} x}{\sqrt {c x^{2} + b} b^{3} \mathrm {sgn}\left (x\right )} - \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} c^{\frac {3}{2}} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} b c^{\frac {3}{2}} + 5 \, b^{2} c^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{3} b^{2} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x}\,{d x} \]________________________________________________________________________________________