Calculus Level 3

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

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

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