(Abridged) We present Herschel/HIFI spectra of the H2O 1113 GHz and H2O+ 1115 GHz lines toward five nearby prototypical starburst/AGN systems, and OH+ 971 GHz spectra toward three of these. The beam size of 20 corresponds to resolutions between 0.35 and 7 kpc. The observed line profiles range from pure absorption (NGC 4945, M82) to P-Cygni indicating outflow (NGC 253, Arp 220) and inverse P-Cygni indicating infall (Cen A). The similarity of the H2O, OH+, and H2O+ profiles to each other and to HI indicates that diffuse and dense gas phases are well mixed. We estimate column densities assuming negligible excitation (for absorption features) and using a non-LTE model (for emission features), adopting calculated collision data for H2O and OH+, and rough estimates for H2O+. Column densities range from ~10^13 to ~10^15 cm^-2 for each species, and are similar between absorption and emission components, indicating that the nuclear region does not contribute much to the emission in these ground-state lines. The N(H2O)/N(H2O+) ratios of 1.4-5.6 indicate an origin of the lines in diffuse gas, and the N(OH+)/N(H2O+) ratios of 1.6-3.1 indicate a low H2 fraction (~11%) in the gas. Adopting recent Galactic values for the average gas density and the ionization efficiency, we find ionization rates for our sample galaxies of ~3x10^-16 s^-1 which are similar to the value for the Galactic disk, but ~10x below that of the Galactic Center and ~100x below estimates for AGN from excited-state H3O+ lines. We conclude that the ground-state lines of water and its associated ions probe primarily non-nuclear gas in the disks of these centrally active galaxies. Our data thus provide evidence for a decrease in ionization rate by a factor of ~10 from the nuclei to the disks of galaxies, as found before for the Milky Way.
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