Just a quick post on how to fix this, I’ve seen numerous people facing this issue on the latest version of SQL Server Data tools and was having it myself. I assume there is some problem with the installer rather than actually with the SSDT, to get it installed you need to install SSDT from Visual Studio and then install the Integration Services, Report Services and Analytics Services parts from the extracted packages.
From control panel modify your version of Visual Studio.
From the individual components tab tick ‘SQL Server Data Tools’
Now open a command prompt and change directory to the download location of the SSDT Installer file that you have downloaded.
Run: SSDT-Setup-enu.exe /layout
This will extract the various installers into a ‘payloads’ folder in the current location
Now simply run each of the tools you want
When you run each one select which instance of Visual studio to install to and click ‘Install’.