\[ \int \frac {x^2}{a+b \sqrt [3]{x}} \, dx \]
Optimal antiderivative \[ -\frac {3 a^{7} x^{\frac {1}{3}}}{b^{8}}+\frac {3 a^{6} x^{\frac {2}{3}}}{2 b^{7}}-\frac {a^{5} x}{b^{6}}+\frac {3 a^{4} x^{\frac {4}{3}}}{4 b^{5}}-\frac {3 a^{3} x^{\frac {5}{3}}}{5 b^{4}}+\frac {a^{2} x^{2}}{2 b^{3}}-\frac {3 a \,x^{\frac {7}{3}}}{7 b^{2}}+\frac {3 x^{\frac {8}{3}}}{8 b}+\frac {3 a^{8} \ln \! \left (a +b \,x^{\frac {1}{3}}\right )}{b^{9}} \]
command
integrate(x**2/(a+b*x**(1/3)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {3 a^{8} \log {\left (1 + \frac {b \sqrt [3]{x}}{a} \right )}}{b^{9}} - \frac {3 a^{7} \sqrt [3]{x}}{b^{8}} + \frac {3 a^{6} x^{\frac {2}{3}}}{2 b^{7}} - \frac {a^{5} x}{b^{6}} + \frac {3 a^{4} x^{\frac {4}{3}}}{4 b^{5}} - \frac {3 a^{3} x^{\frac {5}{3}}}{5 b^{4}} + \frac {a^{2} x^{2}}{2 b^{3}} - \frac {3 a x^{\frac {7}{3}}}{7 b^{2}} + \frac {3 x^{\frac {8}{3}}}{8 b} \]