Let there be a spherically symmetric charge distribution with charge density varying as   $\rho \left(r\right)={\rho }_0(\frac{5}{4}-\frac{r}{R})uptor=R$ and  $\rho \left(r\right)=0$ for r>R  Where r is the distance from the origin. The electric field at a distance r , $(r<R)$ from the origin is given by

Apply shell theorem , the total charge q upto distance r can be calculated 

$dq=4\pi r^{2}dr\rho$ 

$=4\pi r^{2}dr{\rho }_0[\frac{5}{4}-\frac{r}{R}]$

$=4\pi r{\rho }_0[\frac{5}{4}{r}^{2}dr-\frac{r^3}{R}dr]$

$q=\int dq=4\pi {\rho }_0\underset{0}{\overset{r}{\int }}(\frac{5}{4}{r}^{2}dr-\frac{r^2}{R}dr]$

$q=\int dq=4\pi {\rho }_0\underset{0}{\overset{r}{\int }}(\frac{5}{4}{r}^{2}dr-\frac{r^2}{R}dr]$

As electric field Intensity

$E=\frac{kq}{r^2}=\frac{1}{4\pi {\in }_0}\frac{1}{r^2}4\pi {\rho }_0(\frac{5}{4}\left(\frac{r^3}{3}\right)-\frac{r^4}{4R})$

$=\frac{{\rho }_{0r}}{4{\in }_0}[\frac{5}{3}-\frac{r}{R}]$