Integrating infinitely nested radical

Let $y = \sqrt{x+ \sqrt{x+ \sqrt{x+\cdots}}}$ , then we have:

$\begin{aligned} y & = \sqrt{x+y} \\ y^2 & = x + y \\ y^2 - y - x & = 0 \\ \implies y & = \frac {1+\sqrt{1+4x}}2 & \small \color{#3D99F6} \text{Note that }y > 0 \end{aligned}$

Then, we have:

$\begin{aligned} I & = \int_0^3 \sqrt{x+ \sqrt{x+ \sqrt{x+\cdots}}} \ dx \\ & = \int_0^3 y \ dx \\ & = \int_0^3 \frac {1+\sqrt{1+4x}}2 \ dx \\ & = \frac 12 \left[x + \frac 16 \left(1+ 4x\right)^\frac 32 \right]_0^3 \\ & = \frac 12 \left[3 + \frac 16 \left(1+ 12\right)^\frac 32 - \frac 16(1)^\frac 32 \right] \\ & = \frac 1{12}\left(17 + 13\sqrt{13}\right) \end{aligned}$

$\implies a+b+c+d = 12+17+13+13 = \boxed{55}$