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