Let $\cdot$ denote the Euclidean inner product on $\mathbb R^d$. We know that $$\int_{\mathbb R^d} \exp(-0.5x\cdot Ax)\,\mathrm dx = \sqrt{\det(2\pi A^{-1})}.$$
It holds that $x\cdot Ax = \mathrm{trace}(Axx') = \mathrm{vec}(A)\cdot\mathrm{vec}(xx') = \mathrm{vec}(A)\cdot (x\otimes x)$. Define $a := \mathrm{vec}(A)$ and $x^{\otimes 2} := x\otimes x$. Note that $a\in\mathbb R^{d^2}$.
Now the tensor power can readily be generalized to higher powers by setting $x^{\otimes k} = x^{\otimes (k-1)}\otimes x$ for all positive integers $k$ and $x^{\otimes 1} = x$ and $x^{\otimes 0} = 1$. A natural question would be how to evaluate integrals of the form $$\int_{\mathbb R^d}\exp(-0.5c\cdot x^{\otimes p})\,\mathrm dx$$ for some positive integer $p$ and some $c\in\mathbb R^{d^p}$. Are there any references which deal with this sort of problem?