The minimum is irrational?

Algebra Level 5

\[\large \dfrac{c}{\sqrt{ab + 1 - c}} + \dfrac{b}{\sqrt{ca + 1 - b}} + \dfrac{a}{\sqrt{bc + 1 - a}} \]

For positive real numbers \(a\), \(b\), and \(c\), find the minimum value of the expression above, for \( a+b+c =1 \). If the answer is in the form of \( \dfrac{p\sqrt{q}}{r} \), where \( \gcd(p,r) = 1 \) and \( q\) is square free, find \( p+q+r \).

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