The load need not be so very high.

You may need to sample current and voltage at maybe 6-20kHz each with 10-12 bit resolution depending on expected quality at highly distorted curve forms.

You either use RAM and store i and u for maybe 10-100 periods of 50Hz or 60Hz before doing the computations.

Or you have to do 3 multiplications (i*i, u*u, i*u) and a couple of additions for each i+u sample until you have enough samples to compute Urms, Irms, Prms, apparent power, power factor, crest factor. The biggest problem here is microcontrollers that does not have a reasonably fast 8x8 bit => 16 bit multiplication.

For real-world non-linear loads, it will not take many periods of data or sampling frequency or sampling resolution to get better precision than a solution that relies on the phase shift and average or top-level current and voltage. And the amount of samples used will also greatly reduce the errors from measurement noise.

The following is a nice link:
http://dataforth.com/catalog/pdf/an101.pdf

Note at the end, the table for percentage error depending on number of harmonics processed. Note especially what happens if only the base frequency is processed.