\[ \int x^6 \left (a+b x^2\right )^{9/2} \, dx \]
Optimal antiderivative \[ \frac {3 a^{3} x^{7} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{256}+\frac {3 a^{2} x^{7} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{128}+\frac {9 a \,x^{7} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{224}+\frac {x^{7} \left (b \,x^{2}+a \right )^{\frac {9}{2}}}{16}-\frac {45 a^{8} \arctanh \! \left (\frac {x \sqrt {b}}{\sqrt {b \,x^{2}+a}}\right )}{32768 b^{\frac {7}{2}}}+\frac {45 a^{7} x \sqrt {b \,x^{2}+a}}{32768 b^{3}}-\frac {15 a^{6} x^{3} \sqrt {b \,x^{2}+a}}{16384 b^{2}}+\frac {3 a^{5} x^{5} \sqrt {b \,x^{2}+a}}{4096 b}+\frac {9 a^{4} x^{7} \sqrt {b \,x^{2}+a}}{2048} \]
command
integrate(x**6*(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
\[ \frac {45 a^{\frac {15}{2}} x}{32768 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 a^{\frac {13}{2}} x^{3}}{32768 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{\frac {11}{2}} x^{5}}{16384 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {4099 a^{\frac {9}{2}} x^{7}}{28672 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {8191 a^{\frac {7}{2}} b x^{9}}{14336 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {1699 a^{\frac {5}{2}} b^{2} x^{11}}{1792 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {725 a^{\frac {3}{2}} b^{3} x^{13}}{896 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {79 \sqrt {a} b^{4} x^{15}}{224 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {45 a^{8} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{32768 b^{\frac {7}{2}}} + \frac {b^{5} x^{17}}{16 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]