This paper reports a search for triboson (WWW -/+)-W-+/--W-+/- production in two decay channels ((WWW -/+)-W-+/--W-+/- -> l(+/-)vl(+/-)vl(-/+) v and (WWW -/+)-W-+/--W-+/- -> l(+/-)vl(+/-)vjj with l = e, mu) in proton-proton collision data corresponding to an integrated luminosity of 20.3 fb(-1) at a centre-of-mass energy of 8 TeV with the ATLAS detector at the Large Hadron Collider. Events with exactly three charged leptons, or two leptons with the same electric charge in association with two jets, are selected. The total number of events observed in data is consistent with the Standard Model (SM) predictions. The observed 95% confidence level upper limit on the (WWW -/+)-W-+/--W-+/- production cross section is found to be 730 fb with an expected limit of 560 fb in the absence of SM (WWW -/+)-W-+/--W-+/- production. Limits are also set on WWWW anomalous quartic gauge couplings.