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 $a+b+c+d$ .