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Calculus Level 5

\[\displaystyle \large\int_{0}^{\infty}{\frac{dx}{{x}^{2/3}(1+\sqrt{2}x+{x}^{2})}}\]

If the integral above evaluates to \(\displaystyle \frac{{\pi}^{a}\sqrt{b}}{c}\) for positive integers \(a,b,c\) with \(b\) square-free, then submit your answer as \(a+b+c\).

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