  We consider functionals of the form
$$\mathcal{F}(u):=\int_\Omega\!F(x,u,\nabla u)\,\mathrm{d} x,$$ where
$\Omega\subseteq\mathbb{R}^n$ is open and bounded. The integrand
$F\colon\Omega\times\mathbb{R}^N\times\mathbb{R}^{N\times n}\to\mathbb{R}$ is
assumed to satisfy the classical assumptions of a power $p$-growth and the
corresponding strong quasiconvexity. In addition, $F$ is H\"older continuous
with exponent $2\beta\in(0,1)$ in its first two variables uniformly with
respect to the third variable, and bounded below by a quasiconvex function
depending only on $z\in\mathbb{R}^{N\times n}$. We establish that strong local
minimizers of $\mathcal{F}$ are of class $\mathrm{C}^{1,\beta}$ in an open
subset $\Omega_0\subseteq\Omega$ with
$\mathcal{L}^n(\Omega\setminus\Omega_0)=0$. This partial regularity also holds
for a certain class of weak local minimizers at which the second variation is
strongly positive and satisfying a $\mathrm{BMO}$-smallness condition. This
extends the partial regularity result for local minimizers by Kristensen and
Taheri (2003) to the case where the integrand depends also on $u$. Furthermore,
we provide a direct strategy for this result, in contrast to the blow-up
argument used for the case of homogeneous integrands.
