\[ \int \frac {A+B x+C x^2}{x^3 \left (a+b x^2\right )^{9/2}} \, dx \]
Optimal antiderivative \[ \frac {-a \left (\frac {A b}{a}-C \right )-b B x}{7 a^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-13 b B x -14 A b +7 a C}{35 a^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-87 b B x -105 A b +35 a C}{105 a^{4} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\left (9 A b -2 a C \right ) \arctanh \left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )}{2 a^{\frac {11}{2}}}+\frac {-93 b B x -140 A b +35 a C}{35 a^{5} \sqrt {b \,x^{2}+a}}-\frac {A \sqrt {b \,x^{2}+a}}{2 a^{5} x^{2}}-\frac {B \sqrt {b \,x^{2}+a}}{a^{5} x} \]
command
integrate((C*x**2+B*x+A)/x**3/(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 \[ \text {Timed out} \]_____________________________________________________