When the installation of Linkedin app in iPhone gets stuck, try this:
Go to Settings > then tap on iTunes & App Stores.
Now, tap on your Apple ID and then Sign out.
Now, restart your iPhone.
After this, go to the App Store and download the app again.
You would be prompted to Sign in, do it then.
In case you are still not able to install then try downloading any free app (just for testing, which can be deleted later).
In case even after this you are not able to install LinkedIn app alone then contact LinkedIn for further help.
For this, go to App Store > find LinkedIn app > now, tap on Reviews > then tap on App Support.