expr = 0.00441039 (Sqrt[2 π] Abs[-6 + t]^3 + Sqrt[2 π] 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])

FullSimplify[expr]

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] // Simplify

f2 = f /. Sign[u_] -> u/Sqrt[u^2] /. Abs[u_] -> Sqrt[u^2] // Simplify

f3 = Rationalize[f, 0];

f3 /. Sign[u_] -> u/Sqrt[u^2] /. Abs[u_] -> Sqrt[u^2] // FullSimplify

(*   (1/1000000000000)441039 (-(653725/4) ((3 - 2 t)^2)^(3/2) -
      5496525/2 ((5 - 2 t)^2)^(3/2) - 5496525/2 ((7 - 2 t)^2)^(3/2) -
      653725/4 ((9 - 2 t)^2)^(3/2) +
      10000 Sqrt[2 [Pi]] ((-6 + t)^2)^(3/2) -
      187497 ((-5 + t)^2)^(3/2) + 8962680 ((-4 + t)^2)^(3/2) +
      28986600 ((-3 + t)^2)^(3/2) + 8962680 ((-2 + t)^2)^(3/2) -
      187497 ((-1 + t)^2)^(3/2) + 10000 Sqrt[2 [Pi]] (t^2)^(3/2))   *)