Let $p > 1$; what can be said about $$ \int_{0}^{\infty} \left(\,\sqrt[p]{\,{x^{p} + 1}\,} - x\right)^{p}\,\mathrm{d}x\ {\large ?} $$ The condition $p > \phi$ is sufficient for convergence, since asymptotically the integrand is $\mbox{O}\left(x^{-p\left(p -1\right)\,\,\,}\right)$.
The case $p = 2$ is not bad \begin{align} &\int_0^{\infty}(\sqrt{x^2+1}-x)^2\,dx \\[3mm] = &\ \lim_{b\to\infty}\left(\!\left.% \frac{2x^{3}}{3}- \frac{2x^{2}}{3} \sqrt{x^{2} + 1} - \frac{2}{3}\sqrt{x^{2} + 1} + x\right)\right\vert_{\ 0}^{\ b} \\[3mm] = &\ 0 - \frac{-2}{3} = \frac{2}{3}. \end{align} The other antiderivatives seemed to involve the hypergeometric function, so the FTC didn't really pan out, and I couldn't reduce it to a beta integral or some such to evaluate it by hand ( though I did run the first few $p \in \mathbb{N}$ through Wolfram Alpha; no clear pattern ).
Maybe there's a clever transformation I'm missing that would make this problem tractable.