Unbalanced Symmetric Inequality

Algebra Level 5

\[\large \dfrac{ab}{c^2+3}+\dfrac{bc}{a^2+3}+\dfrac{ca}{b^2+3}\]

Given that \(a,b\) and \( c\) are positive reals satisfying \(a+b+c=3\). If the maximum value of the expression above can be expressed as \[\dfrac{A\sqrt{B}-C}{D} \; , \] where \(A,B,C\) and \(D\) are positive integers such that \(\gcd(A,C)=\gcd(A,D)=1\) and \(B\) is square-free, determine \(A+B+C+D\).

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