I deleted my last question, since some of you wanted me to rewrite the question properly. I feel sorry for inconvenience, but please understand that this is the first time I use $\texttt{MathJax}$.
Up to now, I tried every method I know in integration, like substituition, partial fractions, uv-method, etc. But seems like nothing works. I would appreciate to have your help. Thanks.
$$ \int_{0}^{1}\frac{\mathrm{d}x}{\left(x + 1\right) \left[x^{2}\left(1 - x\right)\right]^{1/3}}\,, \qquad\qquad\int_{0}^{1}\frac{\mathrm{d}x}{\left(x^{2} + 1\right) \left[x^{2}\left(1 - x\right)\right]^{1/3}} $$
Comment, but easier to enter as an answer.
Wolfy says:
$\int_0^1 \dfrac{dx}{(x + 1) (x^2 (1 - x))^{1/3}} = \dfrac{2^{2/3} π}{\sqrt{3}} ≈2.87923 $
and
$\int _0^1\dfrac{dx}{(x^2+1)(x^2(1-x))^{1/3}} = \dfrac{(3 + \sqrt{3}) π}{3\cdot 2^{2/3}} ≈3.1217 $