\[\large \int_{0}^{1}\sqrt{x+\sqrt{x^2+1}} \, dx=\dfrac AB \left(1 + \sqrt{\sqrt C - D} \right)\]

If the equation above holds true for positive integers \(A,B, C\) and \(D\) such that \(A\) and \(B\) are coprime, find the value of \(A+B+C+D\).

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