To find \(f(x)\), we need to integrate \(f'(x) = x^{-\frac{3}{2}} + 1\).

The integral of \(x^{-\frac{3}{2}}\) is \(2x^{-\frac{1}{2}}\), and the integral of \(1\) is \(x\).

Thus, \(f(x) = 2x^{\frac{1}{2}} + x + c\), where \(c\) is a constant.

We use the condition \(f(4) = 5\) to find \(c\):

\(5 = 2 \times 4^{\frac{1}{2}} + 4 + c\)

\(5 = 2 \times 2 + 4 + c\)

\(5 = 4 + 4 + c\)

\(5 = 8 + c\)

\(c = 5 - 8 = -3\)

Therefore, \(f(x) = 2x^{\frac{1}{2}} + x - 3\).