\[ \int \frac {A+B x^2+C x^4+D x^6+F x^8}{\left (a+b x^2\right )^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (A \,b^{4}-F \,a^{4}\right ) x}{a \,b^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\left (6 A \,b^{4}+B a \,b^{3}-10 F \,a^{4}\right ) x^{3}}{3 a^{2} b^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\left (24 A \,b^{4}+a \left (4 B \,b^{3}+3 C a \,b^{2}-58 F \,a^{3}\right )\right ) x^{5}}{15 a^{3} b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\left (48 A \,b^{4}+a \left (8 B \,b^{3}+6 C a \,b^{2}+15 D a^{2} b -176 F \,a^{3}\right )\right ) x^{7}}{105 a^{4} b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {F \arctanh \left (\frac {x \sqrt {b}}{\sqrt {b \,x^{2}+a}}\right )}{b^{\frac {9}{2}}} \]
command
integrate((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 \[ \text {Timed out} \]_____________________________________________________