f = 0.00441039 (Sqrt[2 [Pi]] Abs[-6 + t]^3 +
 Sqrt[2 [Pi]] Abs[t]^3 + 18.7497 (-1. + t)^3 Sign[1. - 1. t] +
 130.745 (-1.5 + 1. t)^3 Sign[1.5 - 1. t] -
 896.268 (-2. + t)^3 Sign[2. - 1. t] +
 2198.61 (-2.5 + 1. t)^3 Sign[2.5 - 1. t] -
 2898.66 (-3. + t)^3 Sign[3. - 1. t] +
 2198.61 (-3.5 + 1. t)^3 Sign[3.5 - 1. t] -
 896.268 (-4. + t)^3 Sign[4. - 1. t] +
 130.745 (-4.5 + 1. t)^3 Sign[4.5 - 1. t] +
 18.7497 (-5. + t)^3 Sign[5. - 1. t]);

PiecewiseExpand[f, Reals] // FullSimplify