Given \(b>0 \), and \[ \int_{1}^{\infty}\frac{\sqrt{x+b}}{x^2+b}dx \]

If the minimum value of this integral can be written as \(\displaystyle \frac{a\pi}{b} \) with \( \displaystyle \frac{a}{b} \) a irreducible fraction, evaluate \(a+b\)

Credits: OBM (Brazilian Math Olympiad)

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