\delta) - \alpha \sqrt{R_0}~(3 - 2r) - 2\beta R_0~ (2 -r) - \nonumber \\
& 2\gamma R_0 ~(1 + 2 \ln [R_0] - r \ln [R_0]~) \Big] ~~\end{aligned}$$ and $$\begin{aligned}
C_2  =&  -\frac{R_0^{1-r}}{2(q-r)} \Big[2(2q - \delta +  q \delta) - \alpha \sqrt{R_0}~(3 - 2q) - 2\beta R_0~ (2 -q) - \nonumber \\
& 2\gamma R_0 ~(1 + 2 \ln [R_0] - q \ln [R_0]~) \Big] ~~.\end{aligned}$$ In general the equation of state of eqn.(\[eos\]) will be a function of $H$ and hence time in this case and so it can explain the transition from quintessence ($\omega_R > -1$) to phantom dominated regime ($\omega_R < -1$) as predicted by recent observations [@f100; @f101; @f102]. If we see eqn.(\[fR1\]) we can infer that there is a contribution from $R^{\frac{3}{2}}$. This is interesting from the fact that we can have contributions from fractional powers of $R$. We will discuss this in the later part of this study.

For the quintessence scale factor the future event horizon is at $$\label{lqs}
L = a \int_t^{\infty} \frac{d t}{a} = \frac{1}{h-1} \sqrt{\frac{6h(2h-1)}{R}}$$ with the condition $h>1$. Putting the value of $L$ from eq.(\[lqs\]) in eqn.(\[hd\]) we get the form of energy density as $$\label{rhoq}
\rho_{\Lambda} = \frac{n^2 