Volume Integral over Oscillatory Function

Question

I want to calculate the definite integral $$ I = \int_{0}^1\int_{0}^1\int_{0}^1 \operatorname{m} (x,y,z) e^{i \phi(x,y,z)} dx dy dz$$ where $$ \operatorname{m}(x,y,z) = m_0 + m_1 x + m_2 y + m_3z + m_4xy + m_5xz+ m_6yz+ m_7xyz \quad m _n\in \mathbb{R} $$ and $$ \operatorname{\phi}(x,y,z) = p_0 + p_1 x + p_2 y + p_3z + p_4xy + p_5xz+ p_6yz+ p_7xyz \quad p _n\in \mathbb{R}$$ I know that the first integral can be solved by finding a parameterisation $h_x(p)=\phi^{-1}(\phi(x)+i p)$ and using $$ \int_{0}^x \operatorname{m} (x,y,z) e^{i \phi(x,y,z)} dx = \\e^{i \phi(1,y,z)} \int_0^\infty \operatorname{m} (h_0(\rho),y,z)e^{-p}h^\prime_0(\rho)d\rho - e^{i \phi(0,y,z)}\int_0^\infty \operatorname{m} (h_1(\rho),y,z)e^{-p}h^\prime_1(\rho)d\rho $$ Having done this, i could probably do this 'trick' again on each new integral of the form $\int_{0}^1\int_{0}^1f(y,z)e^{ig(y,z)} dy dz$ and then again to integrate over $z$. But this leads to long expressions that are not manageable by hand or even mathematica. Any advice how to proceed?
I did a brief literature research on oscillatory integrals, which lead to the hint with the parametrisation that i used for the first integral, but i couldn't find a generalisation for volume integrals over oscillating functions.
Any help is greatly appreciated!