Get full class name of an inner class instance?

Question

I have a inner class

public class Outer {
   public class Inner {
      ...
   }
}

and need to get that running

System.assertEquals('Outer.Inner', getClassName(new Outer.Inner()));

I found out that you can get the inner class name, but not the outer class:

System.assertEquals('Inner', String.valueOf(classInstance).substring(0, String.valueOf(classInstance).indexOf(':'));

I could do a SOSL query on all ApexClass' body but that would be somewhat quirky...