  We study cubature formulas for d-dimensional integrals with an arbitrary
symmetric weight function of tensor product form. We present a construction
that yields a high polynomial exactness: for fixed degree l=5 or l=7 and large
dimension, the number of knots is only slightly larger than the lower bound of
M\"oller and much smaller compared to the known constructions.
  We also show, for any odd degree l=2k+1, that the minimal number of points is
almost independent of the weight function. This is also true for the
integration over the (Euclidean) sphere.
