g[x_, y_,
  z_] := (HeavisideTheta[x - X] -
    HeavisideTheta[x - XX]) (HeavisideTheta[y - Y] -
    HeavisideTheta[y - YY]) (HeavisideTheta[z - Z] -
    HeavisideTheta[z - ZZ])
Integrate[
 Integrate[
  Integrate[
   g[x a1 + y b1 + z c1 + (1 - x - y - z) d1,
    x a2 + y b2 +
     z c2 + (1 - x - y - z) d2, (x a3 + y b3 +
      z c3 + (1 - x - y - z) d3)], {z, 0, 1 - y - x},
   Assumptions -> {1 - y - x \[Element] Reals, X \[Element] Reals,
     XX \[Element] Reals, Y \[Element] Reals, YY \[Element] Reals,
     Z \[Element] Reals, ZZ \[Element] Reals, a1 \[Element] Reals,
     a2 \[Element] Reals, a3 \[Element] Reals, b1 \[Element] Reals,
     b2 \[Element] Reals, b3 \[Element] Reals, c1 \[Element] Reals,
     c2 \[Element] Reals, c3 \[Element] Reals, d1 \[Element] Reals,
     d2 \[Element] Reals, d3 \[Element] Reals}], {y, 0, 1 - x},
  Assumptions -> {1 - x \[Element] Reals, , X \[Element] Reals,
    XX \[Element] Reals, Y \[Element] Reals, YY \[Element] Reals,
    Z \[Element] Reals, ZZ \[Element] Reals, a1 \[Element] Reals,
    a2 \[Element] Reals, a3 \[Element] Reals, b1 \[Element] Reals,
    b2 \[Element] Reals, b3 \[Element] Reals, c1 \[Element] Reals,
    c2 \[Element] Reals, c3 \[Element] Reals, d1 \[Element] Reals,
    d2 \[Element] Reals, d3 \[Element] Reals}], {x, 0, 1},
 Assumptions -> {X \[Element] Reals, XX \[Element] Reals,
   Y \[Element] Reals, YY \[Element] Reals, Z \[Element] Reals,
   ZZ \[Element] Reals, a1 \[Element] Reals, a2 \[Element] Reals,
   a3 \[Element] Reals, b1 \[Element] Reals, b2 \[Element] Reals,
   b3 \[Element] Reals, c1 \[Element] Reals, c2 \[Element] Reals,
   c3 \[Element] Reals, d1 \[Element] Reals, d2 \[Element] Reals,
   d3 \[Element] Reals}]