$\:\left(\frac{1}{2}+\:\sqrt{2}\right)^5$

$\left(\begin{matrix}5\\0\end{matrix}\right)\cdot \left(\frac{1}{2}\right)^{5}+\left(\begin{matrix}5\\1\end{matrix}\right)\cdot \left(\frac{1}{2}\right)^{4}\sqrt{2}+2\left(\begin{matrix}5\\2\end{matrix}\right)\cdot \left(\frac{1}{2}\right)^{3}+\left(\begin{matrix}5\\3\end{matrix}\right)\cdot \left(\frac{1}{2}\right)^{2}\sqrt{\left(2\right)^{3}}+2\left(\begin{matrix}5\\4\end{matrix}\right)+\left(\begin{matrix}5\\5\end{matrix}\right)\sqrt{\left(2\right)^{5}}$