Rational Root Theorem won't save you this time

Algebra Level 5

Let \(r_1,r_2\) be the two largest real roots of the polynomial \[P(x)=3x^3-17x+5\sqrt{6}\]

If \(r_1+ r_2\) can be expressed as \[\dfrac{\sqrt{a}+\sqrt{b}}{c}\] for positive integers \(a,b,c\), then what is the smallest possible value of \(a+b+c\)?

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