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