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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The minimum is irrational?

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