  Four decay modes of the $B_c^+$ meson into a $J/\psi$ meson and multiple
charged kaons or pions are studied using proton-proton collision data,
collected with the~LHCb detector at centre-of-mass energies of 7, 8, and 13~TeV
and corresponding to an integrated luminosity of $9$~fb$^{-1}$. The decay
$B_c^+\to J/\psi K^+ K^- \pi^+ \pi^+ \pi^-$ is observed for the first time, and
evidence for the $B_c^+\to J/\psi 4\pi^+ 3\pi^-$ decay is found. The decay
$B_c^+\to J/\psi 3\pi^+ 2\pi^-$ is observedand and the previous observation of
the $B_c^+\to\psi(2S) \pi^+ \pi^+ \pi^-$ decay is confirmed using the $\psi(2S)
\to J/\psi \pi^+ \pi^-$ decay mode. Ratios of the branching fractions of these
four $B_c^+$ decay channels are measured.
