Rational Root Theorem won't save you this time

Algebra Level 5

Let r1,r2r_1,r_2 be the two largest real roots of the polynomial P(x)=3x317x+56P(x)=3x^3-17x+5\sqrt{6}

If r1+r2r_1+ r_2 can be expressed as a+bc\dfrac{\sqrt{a}+\sqrt{b}}{c} for positive integers a,b,ca,b,c, then what is the smallest possible value of a+b+ca+b+c?

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