\[ \int \frac {x^6 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx \]
Optimal antiderivative \[ -\frac {8 b^{2} \left (-7 A c +6 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{105 c^{4} x}+\frac {4 b \left (-7 A c +6 b B \right ) x \sqrt {c \,x^{4}+b \,x^{2}}}{105 c^{3}}-\frac {\left (-7 A c +6 b B \right ) x^{3} \sqrt {c \,x^{4}+b \,x^{2}}}{35 c^{2}}+\frac {B \,x^{5} \sqrt {c \,x^{4}+b \,x^{2}}}{7 c} \]
command
integrate(x^6*(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
\[ \frac {8 \, {\left (6 \, B b^{\frac {7}{2}} - 7 \, A b^{\frac {5}{2}} c\right )} \mathrm {sgn}\left (x\right )}{105 \, c^{4}} - \frac {{\left (B b^{3} - A b^{2} c\right )} \sqrt {c x^{2} + b}}{c^{4} \mathrm {sgn}\left (x\right )} + \frac {15 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} B - 63 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} B b + 105 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B b^{2} + 21 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} A c - 70 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} A b c}{105 \, c^{4} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (B x^{2} + A\right )} x^{6}}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \]________________________________________________________________________________________