What is the easiest (in terms of duration and agony) way to travel between Nagasaki and Fukuoka?

We'll have spent the better part of the month crossing Honshu: Tokyo-Hakone-Tokyo-Minakami-Yamanouchi-Matsumoto-Takayama-Kaga-Kyoto-Nara-Osaka-Matsue-Miyajima-Nagasaki.

Needless to say, by the end of our month in Japan, we'll have gotten out fill of sightseeing and adventure. That said, any further travelling to accommodate an easy return to NRT Int'l would be based entirely on practicality + convenience.

We only have 3 nights booked for Nagasaki and there's a lot to see and do, so I wonder if I should extend our return trip by one night in order to take the train to Fukuoka and add one extra night there? How long is the shinkansen between Nagasaki and Fukuoka?

Thanks so much!!