\[ \int \frac {x^3 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {x^{3} \left (B a -\left (A b -a C \right ) x \right )}{7 a b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {x \left (3 B a +\left (3 A b +4 a C \right ) x \right )}{35 a \,b^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {3 b B x -6 A b -8 a C}{105 a \,b^{3} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 B x}{35 a^{2} b^{2} \sqrt {b \,x^{2}+a}} \]
command
integrate(x**3*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ A \left (\begin {cases} - \frac {2 a}{35 a^{3} b^{2} \sqrt {a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt {a + b x^{2}} + 105 a b^{4} x^{4} \sqrt {a + b x^{2}} + 35 b^{5} x^{6} \sqrt {a + b x^{2}}} - \frac {7 b x^{2}}{35 a^{3} b^{2} \sqrt {a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt {a + b x^{2}} + 105 a b^{4} x^{4} \sqrt {a + b x^{2}} + 35 b^{5} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) + B \left (\frac {7 a x^{5}}{35 a^{\frac {11}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {9}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {7}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 35 a^{\frac {5}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{7}}{35 a^{\frac {11}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {9}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {7}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 35 a^{\frac {5}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + C \left (\begin {cases} - \frac {8 a^{2}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} - \frac {28 a b x^{2}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} - \frac {35 b^{2} x^{4}}{105 a^{3} b^{3} \sqrt {a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt {a + b x^{2}} + 315 a b^{5} x^{4} \sqrt {a + b x^{2}} + 105 b^{6} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________