Let $[ t ]$ denote the largest integer less than or equal to $t$. If
$$\int _ { 0 } ^ { 3 } \left( \left[ x ^ { 2 } \right] + \left[ \frac { x ^ { 2 } } { 2 } \right] \right) \mathrm { d } x = \mathrm { a } + \mathrm { b } \sqrt { 2 } - \sqrt { 3 } - \sqrt { 5 } + \mathrm { c } \sqrt { 6 } - \sqrt { 7 }$$
where $\mathrm { a } , \mathrm { b } , \mathrm { c } \in \mathbf { Z }$, then $\mathrm { a } + \mathrm { b } + \mathrm { c }$ is equal to $\_\_\_\_$