Integrate[
Boole[{ -2 <= 2 x + 7 y <= 2 && -2 <= 3 x + 6 y <= 2}](6 x^2 +
33 x*y + 42 y^2)^4, {x, -∞, ∞}, {y, -∞, ∞}]
Integrate[(6 x^2 + 33 x*y + 42 y^2)^4, {x, y} ∈
ImplicitRegion[{ -2 <= 2 x + 7 y <= 2 && -2 <= 3 x + 6 y <= 2}, {x,
y}]]
sol = Solve[{u == 2 x + 7 y, v == 3 x + 6 y}, {x, y}][[1]];
jacobian =
Abs[Det[With[{u = 2 x + 7 y, v = 3 x + 6 y}, Grad[{u, v}, {x, y}]]]];
Integrate[((6 x^2 + 33 x*y + 42 y^2)^4 /. sol)*1/jacobian, {u, -2,
2}, {v, -2, 2}]
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