\[ \int \frac {A+B x^2+C x^4+D x^6}{x^6 \left (a+b x^2\right )^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {A}{5 a \,x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {12 A b -5 B a}{15 a^{2} x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-24 A \,b^{2}+a \left (10 b B -3 a C \right )}{3 a^{3} x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {\left (192 A \,b^{3}-a \left (80 b^{2} B -24 a b C +3 a^{2} D\right )\right ) x}{21 a^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 \left (192 A \,b^{3}-a \left (80 b^{2} B -24 a b C +3 a^{2} D\right )\right ) x}{35 a^{5} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}-\frac {8 \left (192 A \,b^{3}-a \left (80 b^{2} B -24 a b C +3 a^{2} D\right )\right ) x}{105 a^{6} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {16 \left (192 A \,b^{3}-a \left (80 b^{2} B -24 a b C +3 a^{2} D\right )\right ) x}{105 a^{7} \sqrt {b \,x^{2}+a}} \]
command
integrate((D*x**6+C*x**4+B*x**2+A)/x**6/(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 \[ \text {Timed out} \]_____________________________________________________