Automatic vertical alignement of multirow cells

Automatic vertical alignement of multirow cells

I have this MWE
\documentclass{article}
\usepackage{multirow}
\usepackage{bigstrut}
\setlength{\bigstrutjot}{2pt} % height added by \bigstrut
\usepackage{tabu}
\tabulinesep=1.5pt % ensures some offset between cell content and border
\begin{document}
\begin{tabu}{|c|c|}
\hline
bigstrut \bigstrut & \\\cline{1-1}
ordinary & \multirow{-2}{*}{Multirow} \\\hline
\shortstack{short \\ stack} & \\\cline{1-1}
ordinary & \multirow{-2}{*}{Multirow} \\\hline
ordinary & \\\cline{1-1}
$x=\int_0^\infty\frac{a}{b}$ & \multirow{-2}{*}{Multirow} \\\hline
ordinary & \\\cline{1-1}
$\displaystyle x=\int_0^\infty\frac{a}{b}$ & \multirow{-2}{*}{Multirow}
\\\hline
\end{tabu}
\end{document}
which shows some cases where content of multirow cells is not exactly
where it should be.

I guess a lot of people are familiar with this issue, as I've seen a dozen
of work-arounds. However all of them required some fine-tuning like
[fixup].
I've found a package called makecell, which partially makes what I want.
Still it has flaws. It won't work for multi-line multirow cells and makes
problems with colortbl package I use for bg colour (multirow has to be
defined in the first row).
Do you have any ideas on how automatic vertical alignement of multirow
cell could be achieved?