We square both sides of the equation to get \begin{align*} \sqrt{x+1}&=x\\ x+1&=x^2\\ x^2-x-1&=0\\ \end{align*}We can solve for $ x $ either by completing the square or applying the quadratic formula, which gives us the smaller solution $ x=\dfrac{1-\sqrt{5}}{2} $ and the larger solution $ x=\dfrac{1+\sqrt{5}}{2} $. Consequently, $ a=1 $, $ b=5 $, and $ c=2 $, so $ a+b+c=\boxed{8} $. Note that the larger of these two roots is just the value of $ \phi $, the golden ratio.