\[ \int \frac {x^2 \left (A+B x^2+C x^4+D x^6+F x^8\right )}{\left (a+b x^2\right )^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (A \,b^{4}-a \left (B \,b^{3}-C a \,b^{2}+D a^{2} b -F \,a^{3}\right )\right ) x^{3}}{7 a \,b^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\left (4 A \,b^{4}+a \left (3 B \,b^{3}-10 C a \,b^{2}+17 D a^{2} b -24 F \,a^{3}\right )\right ) x^{3}}{35 a^{2} b^{4} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {\left (8 A \,b^{4}+a \left (6 B \,b^{3}+15 C a \,b^{2}-71 D a^{2} b +162 F \,a^{3}\right )\right ) x^{3}}{105 a^{3} b^{4} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\left (2 D b -9 F a \right ) \arctanh \left (\frac {x \sqrt {b}}{\sqrt {b \,x^{2}+a}}\right )}{2 b^{\frac {11}{2}}}-\frac {\left (D b -4 F a \right ) x}{b^{5} \sqrt {b \,x^{2}+a}}+\frac {F x \sqrt {b \,x^{2}+a}}{2 b^{5}} \]
command
integrate(x**2*(F*x**8+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 \[ \text {Timed out} \]_____________________________________________________