\[ \int \frac {x^8}{\left (a+b x^{3/2}\right )^{2/3}} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{5} \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {1}{3}}}{b^{6}}+\frac {5 a^{4} \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {4}{3}}}{2 b^{6}}-\frac {20 a^{3} \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {7}{3}}}{7 b^{6}}+\frac {2 a^{2} \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {10}{3}}}{b^{6}}-\frac {10 a \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {13}{3}}}{13 b^{6}}+\frac {\left (a +b \,x^{\frac {3}{2}}\right )^{\frac {16}{3}}}{8 b^{6}} \]
command
integrate(x**8/(a+b*x**(3/2))**(2/3),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {729 a^{5} \sqrt [3]{a + b x^{\frac {3}{2}}}}{728 b^{6}} + \frac {243 a^{4} x^{\frac {3}{2}} \sqrt [3]{a + b x^{\frac {3}{2}}}}{728 b^{5}} - \frac {81 a^{3} x^{3} \sqrt [3]{a + b x^{\frac {3}{2}}}}{364 b^{4}} + \frac {9 a^{2} x^{\frac {9}{2}} \sqrt [3]{a + b x^{\frac {3}{2}}}}{52 b^{3}} - \frac {15 a x^{6} \sqrt [3]{a + b x^{\frac {3}{2}}}}{104 b^{2}} + \frac {x^{\frac {15}{2}} \sqrt [3]{a + b x^{\frac {3}{2}}}}{8 b} & \text {for}\: b \neq 0 \\\frac {x^{9}}{9 a^{\frac {2}{3}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________