Instead of numerical integration I recommend a symbolic approach. Since no definitions of $a, b, c, d$ functions are given it can't be resonably answered what you are doing wrong.
We start with a simple observation:
Factor[(14 x^2 + 61 x*y + 42 y^2)^3] /. {7 x + 6 y -> w, 2 x + 7 y -> z}
w^3 z^3
alternatively
Eliminate[{(14 x^2 + 61 x*y + 42 y^2)^3 == int, 7 x + 6 y == w, 2 x + 7 y == z}, {x, y}]
w^3 z^3 == int
Obviously the jacobian of the transformation $(x,y)$ to $(w,z)$ is not singular. Moreover we need to observe that we integrate with respect to $w$ from $-6$ to $6$ as well as with respect to $z$ from $-6$ to $6$. Integrating an antisymmetric integrand $w^3 z^3$ over a symmetric region yields neccesarily $0$. We don't need to calculate the jacobian (this trivial excercise yields $\frac{1}{37}$).
Integrate[(w z)^3, {w, -6, 6}, {z, -6, 6}]
0
This plot illustrates the linear transformation of the variables and contours of the function.
GraphicsRow[ RegionPlot[ ##2, PlotPoints -> 60, Mesh -> 11, MeshFunctions -> #1, ColorFunction -> "StarryNightColors"]& @@@ { {{2 #1 + 7 #2 &, 7 #1 + 6 #2 &}, -6 < 2 x + 7 y < 6 && -6 < 7 x + 6 y < 6, {x, -2.5, 2.5}, {y, -2.5, 2.5}}, {{#1 &, #2 &}, -6 < z < 6 && -6 < w < 6, {w, -10, 10}, {z, -10, 10}}}]
In order to ensure that our approach is correct let's calculate integral over the region where the both varables are positive i.e. {w, 0, 6}, {z, 0, 6}:
Integrate[ w^3 z^3, {w, 0, 6}, {z, 0, 6}]/37
104976/37
% == Integrate[(14 x^2 + 61 x*y + 42 y^2)^3 Boole[ 0 <= 2 x + 7 y <= 6 && 0 <= 7 x + 6 y <= 6], {x, -6, 6}, {y, -6, 6}]
True
Alternatively we could find the integral with
Integrate[ (14 x^2 + 61 x*y + 42 y^2)^3 HeavisideTheta[2 x + 7 y, 6 - 2 x - 7 y, 7 x + 6 y, 6 - 7 x - 6 y], {x, -∞, ∞}, {y, -∞, ∞}]
However the latter method is considerably slower.
