\[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (-5 A c +4 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{15 c^{3} x}-\frac {\left (-5 A c +4 b B \right ) x \sqrt {c \,x^{4}+b \,x^{2}}}{15 c^{2}}+\frac {B \,x^{3} \sqrt {c \,x^{4}+b \,x^{2}}}{5 c} \]
command
integrate(x^4*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (4 \, B b^{\frac {5}{2}} - 5 \, A b^{\frac {3}{2}} c\right )} \mathrm {sgn}\left (x\right )}{15 \, c^{3}} + \frac {{\left (B b^{2} - A b c\right )} \sqrt {c x^{2} + b}}{c^{3} \mathrm {sgn}\left (x\right )} + \frac {3 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} B - 10 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B b + 5 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} A c}{15 \, c^{3} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (B x^{2} + A\right )} x^{4}}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \]________________________________________________________________________________________