I use Persil biological powder as every repair man I have met, for both washing machine and dishwasher, say using powder is better than tablets or liquid for some reason. I add vanish powder, the one in the green tub for extra hygiene and my whites are very white.
I don't ever use tumble dryer for clothes either, not sure if it makes a difference.
liquids do not contain bleach so you'd be better using a powder as they do contain bleach and should help whites to stay white. I also use a scoop of oxi action crystal white but dont know if it helps.