Looped thrice in an integrated Cycle, that's Cryptic

Calculus Level 5

\[\large{\int _{ 0 }^{ 2 }{ \sqrt { x+\sqrt { \frac { { x }^{ 2 } }{ 2 } +\sqrt { \frac { { x }^{ 3 } }{ 3 } +1 } } } dx }=? }\]

Please give answer upto \(5\) decimal places

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