Table[NIntegrate[
  Sqrt[1 - x1^2] Sqrt[1 - x2^2] Sqrt[1 - x3^2]*0.5/(-0.5^2 + 4 (x1 + x2 + x3)^2),
{x1, 0, 1}, {x2, 0, 1}, {x3, 0, 1},
PrecisionGoal -> 6, MinRecursion -> 3,   Method -> rule],
{rule, {"GlobalAdaptive", "MultidimensionalRule", "NewtonCotesRule", "GaussKronrodRule", "ClenshawCurtisRule"}}]