Probability density function (PDF) of a scaled non-central chi squared distribution

Question

I know that:

1. If $X$ is exponentially distributed with a rate parameter $\lambda$, then $cX$ is exponentially distributed with a rate parameter $\frac{\lambda}{c}$.

2. If $X$ is central chi-squared distributed with $\nu$ degree of freedom, then $cX$ is gamma distributed with shape $\frac{\nu}{2}$ and scale parameter $2c$.

where $c$ is a positive constant. My question is:

If $X$ is non-central chi-squared distributed with $\nu$ degree of freedom, then how is $cX$ distributed? What is the pdf of $cX$ ?