Apply shell theorem the total charge upto distance $r$ can be calculated as followed $d q=4 \pi r^2 \cdot d r . \rho$
$$
\begin{aligned}
& =4 \pi r^2 \cdot d r \cdot \rho_0\left[\frac{5}{4}-\frac{r}{R}\right] \\
& =4 \pi \rho_0\left[\frac{5}{4} r^2 d r-\frac{r^3}{R} d r\right] \\
& \int d q=q=4 \pi \rho_0 \int_0^r\left(\frac{5}{4} r^2 d r-\frac{r^3}{R} d r\right) \\
& =4 \pi \rho_0\left[\frac{5}{4} \frac{r^3}{3}-\frac{1}{R} \frac{r^4}{4}\right] \\
& E=\frac{k q}{r^2} \\
& =\frac{1}{4 \pi \varepsilon_0} \frac{1}{r^2} \cdot 4 \pi \rho_0\left[\frac{5}{4}\left(\frac{r^3}{3}\right)-\frac{r^4}{4 R}\right] \\
& E=\frac{\rho_0 r}{4 \varepsilon_0}\left[\frac{5}{3}-\frac{r}{R}\right] \\
&
\end{aligned}
$$