Definite Integration

Let \([t]\) denote the largest integer less than or equal to \(t\). If \(\int_\limits0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) \mathrm{d} x=\mathrm{a}+\mathrm{b} \sqrt{2}-\sqrt{3}-\sqrt{5}+\mathrm{c} \sqrt{6}-\sqrt{7}\), where \(\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathbf{Z}\), then \(\mathrm{a}+\mathrm{b}+\mathrm{c}\) is equal to __________.