Supernova (SN) 1987A was a peculiar H-rich event with a long-rising (LR) light curve (LC), stemming from a compact blue supergiant star (BSG). Only a few similar events have been presented in the literature. We present new data for a sample of 6 LR Type II SNe (SNe II), 3 of which were discovered and observed by the Palomar Transient Factory (PTF) and 3 observed by the Caltech Core-Collapse Project (CCCP). Our aim is to enlarge the family of LR SNe II, characterizing their properties. Spectra, LCs, and host-galaxies (HG) of these SNe are presented. Comparisons with known SN 1987A-like events are shown, with emphasis on the absolute magnitudes, colors, expansion velocities, and HG metallicities. Bolometric properties are derived from the multiband LC. By modeling the early-time LCs with scaling relations derived from the SuperNova Explosion Code (SNEC) models of MESA progenitor stars, we estimate the progenitor radii of these SNe and other progenitor parameters. We present PTF12kso, a LR SN II with the largest amount of 56Ni mass for this class. PTF09gpn and PTF12kso are found at the lowest HG metallicities for this SN group. The variety of early LC luminosities depends on the wide range of progenitor radii, from a few tens of solar radii (SN 2005ci) up to thousands (SN 2004ek) with intermediate cases between 100 (PTF09gpn) and 300 solar radii (SN 2004em). We confirm that LR SNe II with LC shapes closely resembling that of SN 1987A generally arise from BSGs. However, some of them likely have progenitors with larger radii (~300 solar radii, typical of yellow supergiants) and can thus be regarded as intermediate cases between normal SNe IIP and SN 1987A-like SNe. Some extended red supergiant (RSG) stars such as the progenitor of SN 2004ek can also produce LR SNe II if they synthesized a large amount of 56Ni. Low HG metallicity is confirmed as a characteristic of BSG SNe.
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