The Wicked Integral

Calculus Level 5

\[\begin{equation} {\Large\int_0^\infty} \frac{dx}{\sqrt{x} \left[x^2+\left(1+2\sqrt{2}\right)x+1\bigg] \bigg[1-x+x^2-x^3+\cdots+x^{2014}\right]} \end{equation}\]

Given that the integral above is equal to \(\pi(\sqrt a - b) \), where \(a\) and \(b\) are positive integers, find the value of \(a+b\).

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