\[ \int \frac {x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (A -\frac {a \left (b^{2} B -a b C +a^{2} D\right )}{b^{3}}\right ) x^{7}}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\left (b^{2} B -2 a b C +3 a^{2} D\right ) x^{7}}{5 a \,b^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {\left (8 b^{2} B -36 a b C +99 a^{2} D\right ) x^{5}}{60 a \,b^{4} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {D x^{7}}{4 b^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\left (8 b^{2} B -36 a b C +99 a^{2} D\right ) \arctanh \left (\frac {x \sqrt {b}}{\sqrt {b \,x^{2}+a}}\right )}{8 b^{\frac {13}{2}}}+\frac {\left (8 b^{2} B -36 a b C +99 a^{2} D\right ) x^{3}}{12 a \,b^{5} \sqrt {b \,x^{2}+a}}-\frac {\left (8 b^{2} B -36 a b C +99 a^{2} D\right ) x \sqrt {b \,x^{2}+a}}{8 a \,b^{6}} \]
command
integrate(x**6*(D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________