Volume of the $N$-dimensional domain $\sum\limits_{k=1}^N (1 + |x_k|^a)^b\le\varepsilon$ I wish to calculate the following $N$-dimensional integral $$I = \int_0^\infty dx_1 \ldots \int_0^\infty dx_{N} \, H\left(\varepsilon - \sum_{k=1}^N (1 + x_k^a)^b\right),$$ where $a, b$ and $\varepsilon$ are positive reals with $\varepsilon > N$. Also, $H(x)$ is the Heaviside step function.
Is there a general way to attack this kind of integrals?

$$I = \int\limits_0^\infty dx_1 \ldots \int\limits_0^\infty dx_{N} \, H\left(\varepsilon - \sum_{k=1}^N (1 + x_k^a)^b\right)$$
$$ = \int\limits_0^{\large\sqrt[a]{{\sqrt[b]{\varepsilon-N+1}-1}}} dx_1 \int\limits_0^{\large\sqrt[a]{{\sqrt[b]{\varepsilon-(1+x_1^a)^b-N+2}-1}}} dx_2 \ldots \int\limits_0^{\large\sqrt[a]{{\sqrt[b]{\varepsilon-\sum\limits_{k=1}^{N-2}(1+x_k^a)^b-1}-1}}} dx_{N-1}\int\limits_0^{\large\sqrt[a]{{\sqrt[b]{\varepsilon-\sum\limits_{k=1}^{N-1}(1+x_k^a)^b}-1}}} dx_{N}$$
$$ = \int\limits_0^{\large\sqrt[a]{{\sqrt[b]{\varepsilon-N+1}-1}}} dx_1 \int\limits_0^{\large\sqrt[a]{{\sqrt[b]{\varepsilon-(1+x_1^a)^b-N+2}-1}}} dx_2 \ldots \int\limits_0^{\large\sqrt[a]{{\sqrt[b]{\varepsilon-\sum\limits_{k=1}^{N-2}(1+x_k^a)^b-1}-1}}} \large\sqrt[a]{{\sqrt[b]{\varepsilon-\sum\limits_{k=1}^{N-1}(1+x_k^a)^b}-1}}\,dx_{N-1},$$
and the further exact integration seems impossible.
Various substitutions also do not simplify calculations.
