Unfortunately, this generally doesn't work right now in Ubuntu. See: 3+ monitors, nvidia + intel graphics
X isn't configured to load two different vendors' driver packages (in this case, xserver-xorg-video-intel and fglrx or xserverg-xorg-video-ati) simultaneously. There are many posts on this forum asking about this, and it's not likely that Ubuntu will be able to implement this feature the way it is on Windows. I posted a bug report on Launchpad and was told that it may make it into the next version of nouveau, the open-source nVidia driver, but nVidia is the one vendor whose proprietary drivers are almost always preferable, so that doesn't necessarily help very much (and it doesn't help AMD+Intel users at all).
On top of this, most new mid-to-high-end video cards manufactured from 2012 onwards support 4+ displays, so that may decrease the priority of this feature going forward.
You may be able to hack something together that uses two separate instances of X (such that you wouldn't be able to drag applications between the Intel and AMD displays) but it'd require not using XRANDR (meaning you'd lose the Display configuration program in newer versions of Ubuntu, along with several other more user-friendly implementations like hotplugging displays), so I can't really recommend it unless you have years of Xorg.conf editing under your belt. If you're willing to drop a little bit of cash, you'd probably have better luck buying a second cheap AMD card that can be plugged into a spare PCI slot (e.g. http://www.club-3d.com/index.php/products/reader.en/product/radeon-hd-5450-pci-noiseless-edition.html) and using that instead of the Intel graphics, as it will likely work concurrently with the other one.