WinX DVD Ripper Platinum ($46, free feature-limited demo) does a very good job backing up and transcoding your DVD collection, including your copy-protected discs.
Be aware, however, that according to the Digital Millennium Copyright Act (DMCA) this is both legal and illegal. Copying is fine for the purposes of backup, while breaking the copy protection is not. Yes, a conundrum.
It's unlikely that the Feds will bust down your door to haul you in for backing up your disc or converting it for viewing on your phone or tablet. Now if you tried selling a thousand copies, that's another matter.
WinX DVD Ripper Platinum worked flawlessly in my tests. I don't have all the latest DVD movies discs, but it worked with everything I tried. The company, Digiarty, makes a point of calling out their commitment to breaking the latest copy protection schemes, if and when they arrive.
In addition to copying DVDs, WinX DVD Ripper Platinum will automatically convert them to video files: AVI, MP4, MPEG-2, WMV, FLV, MOV, or just a soundtrack. If you prefer, you can select a template for the iPod, Apple TV, iPhone, iPad, HTC devices, Samsung devices, PSP, and 3GP. You may also tweak the audio and video quality for smaller or larger files. I tried several different formats and all played perfectly.
The only negative about WinX DVD Ripper Platinum is the price. There are free DVD rippers out there (such as DVD Shrink) as well as freebies (such as Freemake Video Converter) to turn them into video files. WinX DVD Ripper Platinum does streamline the ripping/converting process so it could be worth the outlay if have a lot of discs to back up or convert.