To find the equation of the curve, we need to integrate the derivative \(\frac{dy}{dx} = \frac{8}{\sqrt{4x + 1}}\).

The integral of \(\frac{8}{\sqrt{4x + 1}}\) is:

\(y = 8 \int (4x + 1)^{-\frac{1}{2}} \, dx\)

Using the substitution \(u = 4x + 1\), \(du = 4 \, dx\), so \(dx = \frac{1}{4} \, du\).

Thus, the integral becomes:

\(y = 8 \int u^{-\frac{1}{2}} \cdot \frac{1}{4} \, du\)

\(y = 2 \int u^{-\frac{1}{2}} \, du\)

\(y = 2 \cdot 2u^{\frac{1}{2}} + c\)

\(y = 4\sqrt{4x + 1} + c\)

Using the point \((2, 5)\) to find \(c\):

\(5 = 4\sqrt{4(2) + 1} + c\)

\(5 = 4\sqrt{9} + c\)

\(5 = 12 + c\)

\(c = -7\)

Therefore, the equation of the curve is \(y = 4\sqrt{4x+1} - 7\).