First, you need to have an Android Device ID that's bound to a Google account. If you already have a device with a Google account on it, you can find them out with an app: Device ID. If you don't have yet, you can add one, get the ID and then do a factory reset to remove the Google account from the device.
Now install Google Chrome on your computer if not yet. Right click on the Google Chrome icon, select properties, add the following line to the end of "Target:":
Note that there is a space before the "--ignore-...".
Close Chrome and open it again. Add the APK Downloader extension. When it asks for options, give the Google account name, ID and password you found out above:
Login and save setting. That's it. You no longer need to do above again.
Now open http://play.google.com, find your app, open the app page but don't click on the "Install" link; instead, an icon of robot head with two antennas will appear on the address bar, click on it and the apk downloading will start.
Once you have the apk file, send it to your phone's SDCARD. On your phone, open a file explorer such as ES File Explorer, find the apk and click on it to install.
Seems a little lengthy, but 99% above is an one-time job. Afterward, you simply get the apk and side load it on your device. The best part is: now you can own the apk forever and you don't need to root your phone to get it. No worry that one day Google might remove the app from the store.
This is the auther's site: