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Answer
$$ \int {2}\pi\frac{{x}^{{1}}}{{3}}{\left(\sqrt{{{1}+\frac{{1}}{{9}}\frac{{x}^{{4}}}{{3}}}}\right)} \, d\,x = -{{\pi\,\left(27\,\ln \left({{\sqrt{x^4+27}-x^2}\over{\sqrt{x^4+27 }+x^2}}\right)-2\,x^2\,\sqrt{x^4+27}\right)}\over{4\,3^{{{5}\over{2 }}}}} $$
Explanation
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