\[ \int \frac {A+B x^2+C x^4+D x^6}{x^4 \left (a+b x^2\right )^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {A}{3 a \,x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {10 A b -3 B a}{3 a^{2} x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\left (80 A \,b^{2}-3 a \left (8 b B -a C \right )\right ) x}{3 a^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\left (160 A \,b^{3}-a \left (48 b^{2} B -6 a b C -a^{2} D\right )\right ) x^{3}}{3 a^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 b \left (160 A \,b^{3}-a \left (48 b^{2} B -6 a b C -a^{2} D\right )\right ) x^{5}}{15 a^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {8 b^{2} \left (160 A \,b^{3}-a \left (48 b^{2} B -6 a b C -a^{2} D\right )\right ) x^{7}}{105 a^{6} \left (b \,x^{2}+a \right )^{\frac {7}{2}}} \]
command
integrate((D*x**6+C*x**4+B*x**2+A)/x**4/(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 \[ \text {Timed out} \]_____________________________________________________