  73 new cubature rules are found for three standard multidimensional integrals
with spherically symmetric regions and weights, using direct search with a
numerical zero-finder. All but four of the new rules have fewer integration
points than known rules of the same degree, and twenty are within three points
of M{\"o}ller's lower bound. Most have all positive coefficients and most have
some symmetry, including some supported by one or two concentric spheres. They
include degree 7 formulas for integration over the sphere and Gaussian-weighted
integrals over all space, each in 6 and 7 dimensions, with 127 and 183 points,
respectively.
