This work presents a motion corrected free breathing Dynamic Contrast Enhanced MRI (DCE-MRI) with improved dynamic contrast reconstruction method called L+S with soft weighting and joint sparsity. Soft weighting achieves accurate motion correction at expense of dynamic contrast. An additional temporal Fast Fourier Transform (FFT) constraint is used to recover the dynamic contrast. A simulated phantom dataset was created with ground truth in this work, enabling quantification of the motion correction and dynamic contrast performance. The proposed method achieved improved dynamic contrast, better motion correction and high reconstruction efficiency simultaneously.