To determine if \(f(x) = (2x - 5)^3 + x\) is increasing, we need to find its derivative \(f'(x)\).

Using the chain rule, the derivative of \((2x - 5)^3\) is \(3(2x - 5)^2 \times 2\).

Thus, \(f'(x) = 3(2x - 5)^2 \times 2 + 1 = 6(2x - 5)^2 + 1\).

We can also express this as \(f'(x) = 24\left(\frac{x - 5}{2}\right)^2 + 1\).

Since \((2x - 5)^2 \geq 0\) for all \(x\), it follows that \(6(2x - 5)^2 + 1 > 0\) for all \(x\).

Therefore, \(f'(x) > 0\) for all \(x \in \mathbb{R}\), which means \(f\) is an increasing function.