I tried to follow your working but I'm afraid I got lost. In these situations, it's best to just forget it and start again.

I'm assuming you want to stick with your substitution so I'm going to try it.

Letting gives

The integral becomes:

using the sum of two cubes formula.

Which you can now integrate term by term.

Edit: I notice, upon reflection, that you already had this exact thing. See my following post.