Here we go, in installation time plus about 5 seconds:

(%i1) solve([x+y+z=0,x^3+y^3+z^3=3,x^5+y^5+z^5=15],[x,y,z]);
(%o1) [[x = - 0.34729635646322, y = 1.879385232208487, z = - 1.532088888888889], [x = - 1.532088888888889, y = 1.879385232208487, z = - 0.34729635646322],
[x = - 0.34729635646322, y = - 1.532088888888889, z = 1.879385232208487], [x = 1.879385232208487, y = - 1.532088888888889, z = - 0.34729635646322],
[x = - 1.532088888888889, y = - 0.34729635646322, z = 1.879385232208487], [x = 1.879385232208487, y = - 0.34729635646322, z = - 1.532088888888889]]


and [x = 1.879385232208487, y = - 1.532088888888889, z = - 0.34729635646322] pretty much matches what I got.