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