Integrating infinitely nested radical

Calculus Level 4

\[\large \int_0^3 \sqrt{x+\sqrt{x+\sqrt{x+\dots}}}\ dx = \frac1a\left(b+c\sqrt{d}\right) \]

If the equation above holds true for some positive integers \(a\), \(b\), \(c\), and \(d\) such that \(\gcd{(a,b,c)}=1\) and \(d\) is square-free, then find the value of \(a+b+c+d\).