\[\large \int_0^{\infty} \dfrac{1}{\sqrt[\leftroot{-2}\uproot{2}3]{x}(x+1)} \, dx =\dfrac{a\pi}{\sqrt{b}}\]

If the equation above holds true for positive integers \(a\) and \(b\) such that \(b\) is square-free, find \(a+b\).

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