\[ \int x^4 \left (a+b x^2\right )^{9/2} \, dx \]
Optimal antiderivative \[ \frac {3 a^{3} x^{5} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{128}+\frac {3 a^{2} x^{5} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{80}+\frac {3 a \,x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{56}+\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {9}{2}}}{14}+\frac {9 a^{7} \arctanh \! \left (\frac {x \sqrt {b}}{\sqrt {b \,x^{2}+a}}\right )}{2048 b^{\frac {5}{2}}}-\frac {9 a^{6} x \sqrt {b \,x^{2}+a}}{2048 b^{2}}+\frac {3 a^{5} x^{3} \sqrt {b \,x^{2}+a}}{1024 b}+\frac {3 a^{4} x^{5} \sqrt {b \,x^{2}+a}}{256} \]
command
integrate(x**4*(b*x**2+a)**(9/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \frac {9 a^{\frac {13}{2}} x}{2048 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{\frac {11}{2}} x^{3}}{2048 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {1027 a^{\frac {9}{2}} x^{5}}{5120 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {6653 a^{\frac {7}{2}} b x^{7}}{8960 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5249 a^{\frac {5}{2}} b^{2} x^{9}}{4480 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {541 a^{\frac {3}{2}} b^{3} x^{11}}{560 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {23 \sqrt {a} b^{4} x^{13}}{56 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {9 a^{7} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2048 b^{\frac {5}{2}}} + \frac {b^{5} x^{15}}{14 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]