Y Y 5 months was a very tricky time with my grumpy DS too .
At that age I used to sit and watch 10 mins or so of Baby TV (Sky channel 625 a few times per day in desperation when I had exhausted all other avenues of entertainment and it usually kept him happy . If you have a baby like this, screw your preconceptions about 'what kind of parent you thought you would be' and just do whatever it takes to minimise the grizzling.
Agree with other posters thought that once your baby can sit alone, crawl and then walk, things will get exponentially easier. DS1 is 16mo now and it is SO much easier dealing with a toddler than a baby who grizzles all the time. DS is still the same charatcer: needs CONSTANT stimulation, but it's much easier nowadays, we go to the park and run around picking up leaves, pushing the buggy around, chasing a ball etc and he is as happy as larry.
I think he just hated being a baby tbh!
I also second swimming: I started a weekly 30 minute 'baby swimming class' at the cheap local council pool when ds was about 5mo and he LOVED it although getting him changed was a stressfull screamfest so maybe go with someone the first few times?
And (man I am going to get double flamed!) have you considered offering a little bit of pureed veg or fruit? Yes, yes, I know the guidelines recommend starting food at 6mo but that's only a few weeks away and not all babies have read the guidelines and it's not likely that every single human will start wanting solid food at exactly the same date, is it?? I started my DS on a little bit of pureed veg once a day at 5mo and he loved it, seemed really ready and enjoyed the experience. It was like he was developmentally ready to start partaking of food, if that makes sense? Might be worth a thought.
Hang in there: I really found this stage the pits. It gets better.