\[ \int \frac {x^6 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-A c +b B \right ) x^{5}}{b c \sqrt {c \,x^{4}+b \,x^{2}}}-\frac {2 \left (-3 A c +4 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3 c^{3} x}+\frac {\left (-3 A c +4 b B \right ) x \sqrt {c \,x^{4}+b \,x^{2}}}{3 b \,c^{2}} \]
command
integrate(x^6*(B*x^2+A)/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (4 \, B b^{2} - 3 \, A b c\right )} \mathrm {sgn}\left (x\right )}{3 \, \sqrt {b} c^{3}} - \frac {B b^{2} - A b c}{\sqrt {c x^{2} + b} c^{3} \mathrm {sgn}\left (x\right )} + \frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}} B c^{6} - 6 \, \sqrt {c x^{2} + b} B b c^{6} + 3 \, \sqrt {c x^{2} + b} A c^{7}}{3 \, c^{9} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (B x^{2} + A\right )} x^{6}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________