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