The minimum is irrational?

Algebra Level 5

cab+1c+bca+1b+abc+1a\large \dfrac{c}{\sqrt{ab + 1 - c}} + \dfrac{b}{\sqrt{ca + 1 - b}} + \dfrac{a}{\sqrt{bc + 1 - a}}

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

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