Integrating infinitely nested radical

Calculus Level 4

03x+x+x+ dx=1a(b+cd)\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 aa, bb, cc, and dd such that gcd(a,b,c)=1\gcd{(a,b,c)}=1 and dd is square-free, then find the value of a+b+c+da+b+c+d.

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