Go along to many Pick-Your-Own farms Gerald and you will find all sorts of cane and berry fruits all grown very closely together.
So just enjoy.
That said...don't plant them so close that their ever reaching root systems get tangled up.
Yes, like Jona878 said, just be careful with the spacing ... the new plants look so small and unobtrusive, that it's easy to plant them too close together. Remember that they are going to "explode"; so give them enough space to do their thing. Plus, they will set more fuit when they get ample sunlight; so less crowding is better for that.
I have always read that you should NOT plant Black Raspberries anywhere near Red Raspberries... as the Black Raspberries can carry a virus that will kill the fruit production or the Red.
But, no problems that I'm aware of between Blackberries and Raspberries.