My iPhone 5 won't let me download certain apps. They show up on itunes but not on my phone, if I download the app from my computer it loads on my phone then immediately disappears. The 'FREE' button is white in the app store for certain apps. PLEASE HELP!

I'm so frustrated and in desperate need of help, I don't know what else to do! Certain apps such as Tumblr (I've previously had the Tumblr app on my phone) and Vine will NOT dowload on my phone. I've tried deleting all apps from both my iPhone and iTunes, I've tried restoring my phone then backing it up, I've tried downloading the app from iTunes while my phone is plugged into my computer, etc. When I search the apps in the app store they show up but the 'FREE' button is white and unclickable, it's very frustrating. When I download it from iTunes while my phone is plugged in I can watch it download, but when it's done it immediately dissapears from the screen and from my phone. When I plug my phone into iTunes it say I have the Tumblr and Vine apps, but I assure you, I do not. I have deleted and tried to add them so many times but it never works. SOMEONE PLEASE HELP!!!