Evaluate.

Question: (Kunihiko Chikaya 's question on facebook Trợ Giúp Toán Học)
Evaluate $I=\displaystyle\int\limits_0^1 {\frac{x}{{\left[ {\left( {2x - 1} \right)\sqrt {{x^2} + x + 1}  + \left( {2x + 1} \right)\sqrt {{x^2} - x + 1} } \right]\sqrt {{x^4} + {x^2} + 1} }}dx}$
Answer: