First observations of the rare decays $B^+\!\rightarrow K^+\pi^+\pi^-\mu^+\mu^-$ and $B^+\!\rightarrow\phi K^+\mu^+\mu^-$

First observations of the rare decays $B^+\rightarrow K^+\pi^+\pi^-\mu^+\mu^-$ and $B^+\rightarrow \phi K^+\mu^+\mu^-$ are presented using data corresponding to an integrated luminosity of $3.0\,\mbox{fb}^{-1}$, collected by the LHCb experiment at centre-of-mass energies of $7$ and $8\mathrm{\,TeV}$. The branching fractions of the decays are \begin{eqnarray*} \mathcal{B}(B^+\rightarrow K^+\pi^+\pi^-\mu^+\mu^-) &=& (4.36\,^{+0.29}_{-0.27}\,\mathrm{(stat)}\pm 0.21\,\mathrm{(syst)}\pm0.18\,\mathrm{(norm)})\times10^{-7},\\ \mathcal{B}(B^+\rightarrow\phi K^+\mu^+\mu^-) &=& (0.82 \,^{+0.19}_{-0.17}\,\mathrm{(stat)}\,^{+0.10}_{-0.04}\,\mathrm{(syst)}\pm0.27\,\mathrm{(norm)}) \times10^{-7},\end{eqnarray*} where the uncertainties are statistical, systematic, and due to the uncertainty on the branching fractions of the normalisation modes. A measurement of the differential branching fraction in bins of the invariant mass squared of the dimuon system is also presented for the decay $B^+\rightarrow K^+\pi^+\pi^-\mu^{+}\mu^{-}$.