Would you marry someone after 5 months? No, of course you wouldn't. But if you're going to move in with someone and you already have DC it's essentially the same level of commitment. Higher, really, because it's higher stress and in at the deep end.
There is no way you get to this point before a year and even this is much, much too fast. A minimum of two years, I think at the very least.
You need to have spent extended periods of time with each other not on "best behaviour".
You need to be okay with him being a second parent to the DC unless they are older (10+) and/or still have a strong relationship with their dad.
Either way you need to know they get on, including crunch points like discipline, bad behaviour, bad moods.
You need to know how you're going to split and manage responsibilities like housework, childcare, financial support of the household, managing available finances. Never assume. Talk about it.
You need to know how you cope as a couple with stressful situations, emotional decisions, and families blending. (At five months, it seems unlikely that you would have even thrown a joint BBQ).
You need to be on the same page regarding potential future children and/or pets, and if you want more children, to be sensitive to the needs of your older children.
You don't get to have a "trial run" like you would pre DC. You can't move him in and then out again. You need to be 100% sure and then wait 50% of that time again and see if you're still sure. I don't personally have a problem with introducing a new partner to DC early on as long as it's done in a controlled way but moving in is a totally different story, and has such huge consequences. I think it's a really important one to get right. You can't wait too long - you can absolutely move in too soon. In fact, DH and I initially moved in together a year in despite my misgivings and we probably would have split up if he hadn't happened to get a job abroad which took him right away for over a year. It was really lucky (and also could have split us up, TBH, it was so close) - I think if that freak chance hadn't happened, we probably wouldn't still be together.
Be bloody careful. A whirlwind relationship which "just feels right" is a huge red flag, because someone who is emotionally stable is normally keeping things slow at this stage and testing the waters, not jumping in head first. Be careful that this is not a manipulative tactic. Remember when something seems too good to be true it usually is.
Good luck!