If ${\int }_0^1 \left({\mathrm{x}}^{21}+{\mathrm{x}}^{14}+{\mathrm{x}}^7\right){\left(2{\mathrm{x}}^{14}+3{\mathrm{x}}^7+6\right)}^{1/7}\mathrm{dx}=\frac{1}{\mathrm{l}}(11{)}^{\mathrm{m}/\mathrm{n}}$ where $l, m, n\in \mathrm{\mathbb{N}}$, m and n are coprime then$l+m+n$ is equal to ______ 

$\begin{array}{l}{\int }_0^1\left(x^{20}+x^{13}+x^6\right){\left(2x^{21}+3x^{14}+6x^7\right)}^{1/7}dx\\ put 2{\mathrm{x}}^{21}+3{\mathrm{x}}^{14}+6{\mathrm{x}}^7=\mathrm{t}\\ \Rightarrow 42\left({\mathrm{x}}^{20}+{\mathrm{x}}^{13}+{\mathrm{x}}^6\right)\mathrm{dx}=\mathrm{dt}\\ \frac{1}{42}{\int }_0^{11} t^{\frac{1}{7}}dt={\left(\frac{t^{\frac{8}{7}}}{\frac{8}{7}}\times \frac{1}{42}\right)}_0^{11}\\ =\frac{1}{48}{\left(t^{\frac{8}{7}}\right)}_0^{11}=\frac{1}{48}(11{)}^{8/7}\\ l=48,\mathrm{m}=8,\mathrm{n}=7\\ l+\mathrm{m}+\mathrm{n}=63\end{array}$