Area Under a Composite Function

Let $g(x) = \cos x^2$, $f(x) = \sqrt{x}$, and $\alpha$, $\beta$ ($\alpha < \beta$) be the roots of the quadratic equation $18x^2 - 9\pi x + \pi^2 = 0$. What is the area (in sq. units) bounded by the curve $y = (g \circ f)(x)$ and lines $x = \alpha$, $x = \beta$, and $y = 0$? The answer is in the form $\frac{-a+\sqrt{b}}{c}$ for where $a$, $b$, $c$ are positive integers and $\gcd(a,c) = 1$. Input the answer as the sum $a+b+c$.