\[ \int \frac {1}{x^2 \left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {15 c^{2} \arctanh \left (\frac {x \sqrt {b}}{\sqrt {c \,x^{4}+b \,x^{2}}}\right )}{8 b^{\frac {7}{2}}}+\frac {1}{b \,x^{3} \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {5 \sqrt {c \,x^{4}+b \,x^{2}}}{4 b^{2} x^{5}}+\frac {15 c \sqrt {c \,x^{4}+b \,x^{2}}}{8 b^{3} x^{3}} \]
command
integrate(1/x^2/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {15 \, c^{2} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{8 \, \sqrt {-b} b^{3} \mathrm {sgn}\left (x\right )} + \frac {c^{2}}{\sqrt {c x^{2} + b} b^{3} \mathrm {sgn}\left (x\right )} + \frac {7 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} c^{2} - 9 \, \sqrt {c x^{2} + b} b c^{2}}{8 \, b^{3} c^{2} x^{4} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]________________________________________________________________________________________