\[ \int \frac {x^2}{\left (a+b \sqrt [3]{x}\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {3 a^{8}}{b^{9} \left (a +b \,x^{\frac {1}{3}}\right )}+\frac {21 a^{6} x^{\frac {1}{3}}}{b^{8}}-\frac {9 a^{5} x^{\frac {2}{3}}}{b^{7}}+\frac {5 a^{4} x}{b^{6}}-\frac {3 a^{3} x^{\frac {4}{3}}}{b^{5}}+\frac {9 a^{2} x^{\frac {5}{3}}}{5 b^{4}}-\frac {a \,x^{2}}{b^{3}}+\frac {3 x^{\frac {7}{3}}}{7 b^{2}}-\frac {24 a^{7} \ln \! \left (a +b \,x^{\frac {1}{3}}\right )}{b^{9}} \]
command
integrate(x**2/(a+b*x**(1/3))**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \frac {840 a^{8} x^{\frac {176}{3}} \log {\left (1 + \frac {b \sqrt [3]{x}}{a} \right )}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} - \frac {840 a^{7} b x^{59} \log {\left (1 + \frac {b \sqrt [3]{x}}{a} \right )}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} + \frac {840 a^{7} b x^{59}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} + \frac {420 a^{6} b^{2} x^{\frac {178}{3}}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} - \frac {140 a^{5} b^{3} x^{\frac {179}{3}}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} + \frac {70 a^{4} b^{4} x^{60}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} - \frac {42 a^{3} b^{5} x^{\frac {181}{3}}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} + \frac {28 a^{2} b^{6} x^{\frac {182}{3}}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} - \frac {20 a b^{7} x^{61}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} + \frac {15 b^{8} x^{\frac {184}{3}}}{35 a b^{9} x^{\frac {176}{3}} + 35 b^{10} x^{59}} \]