Integral Divisibility

Given that \(a\) and \(b\) are positive integers with \( (a_1,b_1), (a_2,b_2),\ldots,(a_n,b_n) \) as solutions such that \(\dfrac{a^2+b}{b^2-a}\) and \(\dfrac{b^2+a}{a^2-b}\) are both integers, find \[ \sum_{i=1}^{n} a_i b_i. \]

This problem is from Indian International Mathematical Olympiad Training Camp 2015
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