\[ \int \frac {\sqrt {x}}{-\frac {1}{\sqrt [3]{x}}+\sqrt {x}} \, dx \]
Optimal antiderivative \[ 6 x^{\frac {1}{6}}+x +\frac {6 \ln \! \left (1-x^{\frac {1}{6}}\right )}{5}-\frac {3 \ln \! \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}-x^{\frac {1}{6}} \sqrt {5}\right ) \left (-\sqrt {5}+1\right )}{10}-\frac {3 \ln \! \left (2+x^{\frac {1}{6}}+2 x^{\frac {1}{3}}+x^{\frac {1}{6}} \sqrt {5}\right ) \left (\sqrt {5}+1\right )}{10}-\frac {3 \arctan \! \left (\frac {\left (1+4 x^{\frac {1}{6}}+\sqrt {5}\right ) \sqrt {50+10 \sqrt {5}}}{20}\right ) \sqrt {10-2 \sqrt {5}}}{5}-\frac {3 \arctan \! \left (\frac {1+4 x^{\frac {1}{6}}-\sqrt {5}}{\sqrt {10+2 \sqrt {5}}}\right ) \sqrt {10+2 \sqrt {5}}}{5} \]
command
integrate(x**(1/2)/(-1/x**(1/3)+x**(1/2)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \int \frac {x^{\frac {5}{6}}}{\left (\sqrt [6]{x} - 1\right ) \left (\sqrt [6]{x} + x^{\frac {2}{3}} + \sqrt [3]{x} + \sqrt {x} + 1\right )}\, dx \]
Sympy 1.8 under Python 3.8.8 output
\[ 6 \sqrt [6]{x} + x + \frac {6 \log {\left (\sqrt [6]{x} - 1 \right )}}{5} - \frac {3 \sqrt {5} \log {\left (8 \sqrt [6]{x} + 8 \sqrt {5} \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{10} - \frac {3 \log {\left (8 \sqrt [6]{x} + 8 \sqrt {5} \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{10} - \frac {3 \log {\left (- 8 \sqrt {5} \sqrt [6]{x} + 8 \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{10} + \frac {3 \sqrt {5} \log {\left (- 8 \sqrt {5} \sqrt [6]{x} + 8 \sqrt [6]{x} + 16 \sqrt [3]{x} + 16 \right )}}{10} - \frac {3 \sqrt {2} \sqrt {5 - \sqrt {5}} \operatorname {atan}{\left (\frac {2 \sqrt {2} \sqrt [6]{x}}{\sqrt {5 - \sqrt {5}}} + \frac {\sqrt {2}}{2 \sqrt {5 - \sqrt {5}}} + \frac {\sqrt {10}}{2 \sqrt {5 - \sqrt {5}}} \right )}}{5} - \frac {3 \sqrt {2} \sqrt {\sqrt {5} + 5} \operatorname {atan}{\left (\frac {2 \sqrt {2} \sqrt [6]{x}}{\sqrt {\sqrt {5} + 5}} - \frac {\sqrt {10}}{2 \sqrt {\sqrt {5} + 5}} + \frac {\sqrt {2}}{2 \sqrt {\sqrt {5} + 5}} \right )}}{5} \]