\[ \int \frac {x^8 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-A c +b B \right ) x^{7}}{b c \sqrt {c \,x^{4}+b \,x^{2}}}+\frac {8 b \left (-5 A c +6 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{15 c^{4} x}-\frac {4 \left (-5 A c +6 b B \right ) x \sqrt {c \,x^{4}+b \,x^{2}}}{15 c^{3}}+\frac {\left (-5 A c +6 b B \right ) x^{3} \sqrt {c \,x^{4}+b \,x^{2}}}{5 b \,c^{2}} \]
command
integrate(x^8*(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
\[ -\frac {8 \, {\left (6 \, B b^{3} - 5 \, A b^{2} c\right )} \mathrm {sgn}\left (x\right )}{15 \, \sqrt {b} c^{4}} + \frac {B b^{3} - A b^{2} c}{\sqrt {c x^{2} + b} c^{4} \mathrm {sgn}\left (x\right )} + \frac {3 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} B c^{16} - 15 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B b c^{16} + 45 \, \sqrt {c x^{2} + b} B b^{2} c^{16} + 5 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} A c^{17} - 30 \, \sqrt {c x^{2} + b} A b c^{17}}{15 \, c^{20} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (B x^{2} + A\right )} x^{8}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________