A really cheap and simple option worth trying is an indoor dipole. ? Check out this page, , and omit the 20m radials since you're only using it for the bitx40.? I have something similar installed in my house... although I haven't used it much (I spend too much time doing homework right now... not enough playing with radios), I was able to hear stuff on it when I first set it up. ?
You'd need some kind of swr meter to trim it to the correct length, but that's it. ??