The inequality in question is a particular case (with $v=u+1$) of the inequality 
$$\int_u^v x^p\, dx \le(v-u)u^{p/2}v^{p/2}\tag{1}$$
for $v\ge u>0$, where without loss of generality (wlog) $p\in(-1,0]$. By the homogeneity in $(u,v)$, wlog $u=1$, and then (1) can be rewritten as 
$$g(v):=g_p(v):=v^{p+1}-1-(p+1)(v-1) v^{p/2}\le0\tag{2}$$
for $v\ge1$. 

Note that  
$$g'(v)=\tfrac12\, (p+1) v^{p/2-1}h_v(p),$$
where
$$h_v(p):=2 v (v^{p/2}-1)+p-p v,$$
so that $h_v(p)$ is convex in $p$. Also, $h_v(0)=0$ and $h_v(-1)=-(\sqrt v-1)^2\le0$. So, $h_v\le0$ and hence $g'\le0$, which implies that $g$ is decreasing, from $g(1)=0$. Thus, (2) follows.