Using the substitution $u=\sqrt{3x+1}$, show that

$${\int }_1^5\sqrt{12x+4}{e}^{\sqrt{3x+1}}dx$$

may be expressed in the form

$${\int }_a^{b}k{u}^2{e}^{u}du$$

where $a$, $b$ and $k$ are constants to be found.

Hence find, by algebraic integration, the exact value of

$${\int }_1^5\sqrt{12x+4}{e}^{\sqrt{3x+1}}dx$$

giving your answer in simplest form.