}{\partial t}
\int_0^t (t-s)^{-{\beta_2}} \int_0^s
h(a, u(a,x))
dW_ads\\
&=
\frac{1}{\Gamma (2-{\beta_2})}\frac{\partial }{\partial t}
 \int_0^t
 (t-s)^{1-{\beta_2}}
h(s, u(s,x))
dW_s.\end{aligned}$$ Thus can be rewritten as the following stochastic partial differential equation involving fractional time-derivative $$\label{e:1.4}
\partial^{\beta_1}_t u = {\rm div}  \, \vec F + \partial^{\beta_2}_t \int_0^t
h(s, u(s,x)) dW_s.$$ It is this type of stochastic equations and its natural extensions that will be studied in this paper.

Now let $(\Omega,{\mathcal{F}},{\mathbb{P}})$ be a complete probability space, $\{{\mathcal{F}}_{t},t\geq0\}$ be an increasing filtration of $\sigma$-fields ${\mathcal{F}}_{t}\subset{\mathcal{F}}$, each of which contains all $({\mathcal{F}},{\mathbb{P}})$-null sets. We assume that on $\Omega$ we are given an independent family of one-dimensional Wiener processes $W^{1}_{t}$, $W^{2}_{t},...$ relative to the filtration $\{{\mathcal{F}}_{t},t\geq0\}$.

Motivated by , in this paper we consider the following quasi-linear SPDEs of the non-divergence form type $$\begin{aligned}
    \nonumber
  \partial^{\beta}_t u&=&\left(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu+f(u) \right)\\
   && +\sum_{k=1}^\infty\partial^{\gamma}_t \int^t_0 (\sigma^{ijk}u_{x^ix^j} +\mu^{ik}u_{x^i}+\nu^ku+g^k(u))\, d
   W^k_s
    \label{eqn 2014.1}
 \end{aligned}$$ as well as of the divergence form type $$\begin{aligned}
                  \nonumber
  \partial^{\beta}_t u &=&\left( D_i\left(
a^{ij}u_{x^j}+b^iu+f^i(u)\right)+cu+h(u)
  \right) \\
   && + \sum_{k=1}^\infty\partial^{\gamma}_t \int^t_0 (\sigma^{ijk}u_{x^ix^j} +\mu^{ik}u_{x^i}+\nu^ku+g^k(u))\,
    dW^k_s,                  \label{2014.2}
 \end{aligned}$$ given for $\omega\in \Omega, \, t\geq0$ and $x\in {\mathbb{R}}^d$, and study the $L_2$-theory of the equations. The constants $\beta,\gamma\in (0,1)$ are assumed to satisfy the condition $$\label{e:1.7}
\gamma<\beta+1/2.$$ The indices $i$ and 