I need an asymptotic equality for the integral $$\int_{(0,1)^5} \left(\frac{x(1-x)y(1-y)u(1-u)v(1-v)w(1-w)}{1-(1-xyuv)w}\right)^n \frac{dxdydudvdw}{1-(1-xyuv)w} $$ where $$\int_{(0,1)^5}=\int_{0}^1\int_{0}^1\int_{0}^1\int_{0}^1\int_{0}^1$$
Define $$I_n:=\int_{(0,1)^5} \left(\frac{x(1-x)y(1-y)u(1-u)v(1-v)w(1-w)}{1-(1-xyuv)w}\right)^n \frac{dxdydudvdw}{1-(1-xyuv)w} $$ I tried using Laplace's method and so we use the substitution $x_1=2x-1,y_1=2y-1,u_1=2u-1,v_1=2v-1,w_1=2w-1$ to get $$I_n=\frac{1}{2^{5n}}\int_{(-1,1)^5} \left(\frac{(1-x_1^2)(1-y_1^2)(1-u_1^2)(1-v_1^2)(1-w_1^2)}{2^5-(2^4-(x_1+1)(y_1+1)(u_1+1)(v_1+1))(w_1+1) }\right)^n \frac{dx_1dy_1du_1dv_1dw_1}{2^5-(2^4-(x_1+1)(y_1+1)(u_1+1)(v_1+1))(w_1+1))} $$ Any help would be highly appreciated. Thank you.
Edit $1$ Can we use the result from p.$331$ in this book Asymptotic expansions of integrals? I don't know how to apply this result in our question.
Edit $2$ $I_n$ can be written as $$I_n=\int_{(0,1)^5} \exp\{n\phi(x,y,u,v,w)\}g_0(x,y,u,v,w)\ dxdydudvdw$$ where $$\phi(x,y,u,v,w)= \log\left(\frac{x(1-x)y(1-y)u(1-u)v(1-v)w(1-w)}{1-(1-xyuv)w}\right)$$ and $$g_0(x,y,u,v,w)=\frac{1}{1-(1-xyuv)w}$$ So $I_n$ is an integral of Laplace's type.