We will show that the space contains isomorphically the space $l_1$ therefore the space  is not reflexive. We start with the following that I  posed as a question in a previous comment.  
    
  **Fact 1:**  For every $0<\delta < 1 $ $lim_n \frac{\int_1^{1+\delta} n^{1/p} dp } {\int_1^2  n^{1/p} dp } = 1$.  
  
**Proof:**  Since the function $n^{1/p}$, $1\leq p \leq 2$ is decreasing we have that  $\int_1^{1+\delta} n^{1/p} dp >\delta n^{1+\delta}$ and  
 $n^ {1/1+2\delta} > \int_{1+2\delta}^2 n^{1/p} dp$   
 Now for $n\in N$ we have that $ \frac{\int_{1+2\delta}^2 n^{1/p}dp} {\int_1^2  n^{1/p} dp } <   \frac{\int_{1+2\delta}^2 n^{1/p}dp} {\int_1^{1+\delta}  n^{1/p} dp } < \frac{n^{1/1+2\delta }} {\delta n^{1/1+ \delta }} = \frac{1} { \delta} \frac {1}{n^{\delta/ (1+\delta)(1+2\delta)}}$.  
 Hence  for every $0<\delta <1/2$  
 
$lim_n  \frac{\int_{1+2\delta}^2 n^{1/p}dp} {\int_1^2  n^{1/p} dp } =0$ which finishes the proof of Fact 1.  
  
 **Fact 2:**  We start with the following classical result.  
If $(f_n)_n$ is a normalized sequence in $L^1 [1,2]$ which is not uniformly integrable (i.e. there exists $\epsilon>0$ such that for every $\delta>0$ 
there exists a Borel set $A$ with $\lambda (A)<\delta$ and $\int_A |f_n| >  \epsilon$ for infinite  
 
 $n\in N$ ) then $(f_n)_n$ has a subsequnce equivalent to $l_1$ basis.  
This result is due to Kadec and Pelczynski ( see **J. Diestel: Sequences and Series in Banach Spaces (Graduate Texts in Mathematics, 92) p. 93**).  
Next in the space we denote $e_i$ the basis of $l_1$ which is a symmetric basis for the space.For $n\in N $ we set $z_n = \sum_ {i=1} ^{n} e_i$ and $x_n = \frac{1} {\int_1^2  n^{1/p} dp } z_n$ which has norm 1.  
 Consider the function $f_n(p) = |x_n|_p $ $1\leq p \leq 2 $ and Fact 1 yields that the sequence $(f_n)$  it is not uniformly integrable.Therefore $(f_n)$ has a subsequence equivalent to $l_1$ basis which implies that $(x_n)$ satisfies the same property in the norm of the space.  
  I have two questions related to this result.  
  
  
  **Question 1 :** Does the space contain a complemented subspace isomorphic to $l_1$ ?  
  
**Question 2:** Is the space $l_1$ saturated?