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We want to apply Vieta's formulas, but the given equation is not a polynomial equation, because of the $\frac{1}{x}$ term. To turn this equation into an equivalent polynomial equation, we multiply both sides by $x$ and rearrange: $$\begin{array}{rl}1+5x^3& =6x^2-24x\\ 5x^3-6x^2+24x+1& =0.\end{array}$$Now we can use Vieta: the sum of the roots is $a+b+c=\boxed{\frac{6}{5}}.$