Have I broken my IPhone 6 plus?

My phone fell from the top of my chest of drawers on to the (carpet tiled) floor, while in it's silicone case,

There was no screen damage and it seems to turn on and work fine?

This is my first iPhone, so I don't whether it's okay too leave it or whether I should take it in to be looked at?

If it's working I wouldn't worry.

Only way to tell if it was damaged is to watch it for the next couple of days. If everything works, you are good. If you have problems, get it looked at. None of us can guess whether it was damaged or not.

If nothing is cracked and it works fine, then just use it and don't worry.

Looks like its fine. I wouldn't worry.

Yes

If it doesn't look broken and isn't acting broken I would guess it isn't broken.

The iPhone 6 and 6 Plus are susceptible to logic board damage. Most commonly, they get what's referred to as Touch Disease. The symptoms are that the phone will tend to stop accepting touch input, or the phone will input false touches (ghost touch). I'd be careful dropping it often and never put it in your back pocket (so it doesn't bend). If the phone appears to function normally, there isn't any need to take it to a repair shop. They'll just charge you a diagnostic fee and tell you it's fine. Put it in a case with good corner protection and maybe add a glass screen protector if you are a bit clumsy.

Yeah spend your money and take it to get it checked out even though it's working fine

I still have my iPhone 7+ and I only broke the camera lens part, the camera still works though.

Yeah.