I have never thought about the problem using a current source. Were it not for the resistor, the charge would be linear but, of course, the resistor is taking increasing amounts of current as the capacitor takes charge.
So, AFAICT, the equation is still related to 1-e-t/RC. So, at 5 time constants, the capacitor is about 99% charged.
I have attached an LTspice file and a screen shot - the time constant is 1 ms so at 5 ms we're pretty close to fully charged.