the indention here is a bit wild - looks like 8 spaces. Or do you use tabs?

It's recommended and encouraged to use two spaces white space (non tab) indention in Ruby. Whitespaces are preferred over tabs because when someone has different tab settings code just starts to look weird, especially if whitespaces and tabs are mixed.

Would be glad if you could fix that up :-)

Using describe&it with parentheses is highly unusual - it's better style to leave the parentheses out here - I think I've never seen it with them... e.g. describe '#index' do etc.

For consistency and update reasons you could migrate this to the expect syntax as well, e.g. expect(response).to redirect_to(root_path) should work :)

When it is better to use expect syntax?
Thanks for the feedback!

Well expect is the new default for RSpec 3.0+ (and maybe even 2.14). So it is generally encouraged, the old syntax will still be maintained but expect is generally the way to go (among other things with expect you don't need to have monkey patching). Also it helps in some edge cases, see this: http://myronmars.to/n/dev-blog/2012/06/rspecs-new-expectation-syntax