Honestly, if I were you I would let baby stay as long as you can, and that you'er comfortable with. I was induced at 38+3 with DD due to what they assumed was my waters breaking (turned out to be hind waters and I didn't necessarily need to be induced) I was put on a drip straight away. I found it really horrible. I was monitored the whole time so was on my back on the bed. I was in labour for 23hours. My cervix was not favourable and it was mentioned many times for me to either have a c/s or vacuume extraction. Baby was very slow to come down and in my opinion was not ready to be born. I was lucky she did come down and I managed to push (for 3hours!) her out vaginally with no help except a 3rd degree epesiotomy. I also ended up having 2 shots of pethadine (even though they made me vomit and heave for about an hour after having the shots) and gas for a little while until it made me sick too. So I was really out of it when DD was born. I didn't want her to have any formula but was just so exhausted and off my face, I let them give her a bottle. I blame the huge issues and truama I suffered from her birth for us not being able to bf well.
I have also had a totally natural birth with my DS at 41+3 and I am praying with my all that I have another birth just like his this time. It was wonderful and I was on such a natural high after he was born. I remember everything and bf him a few mins after his birth. It was just devine.

**This is totally my own opinion and experience. Yours may be a delightful experience if you are induced. There a re several ways to be induced and it can be done slowly, unlike mine which was going from no contractions to every 3 or so mins in about 15mins. Many women have just as lovely an induced birth as I had with my natural birth.

Good luck hun.