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I am doing this work by hand, but it takes a lot of time and I make several calculation errors, so I was thinking to make Mathematica to calculate this for me, but I am stuck at the very beginning.

I am working with tensors like this:

$XXV_ijk = dfrac16(X_iX_jV_k+X_iX_kV_j+X_jX_iV_k+X_kX_jV_i+X_jX_kV_i+X_kX_iV_j)$
$-dfrac15(delta_ij, (Xcdot X) V_k+delta_ik (Xcdot V) X_j+delta_jk (Xcdot V) X_i)$

where $X_i$ and $V_i$ are the components of the 3-vectors $vecX$ and $vecV$. I have to multiply these tensors, for example

$XXV times XXV = dfrac225V^2+dfrac825 (Xcdot V)^2$

I think I can obtain the first part with Tuples and Total(?) but I don't know how to obtain the part with the Kroeneker deltas; if I can write these tensors correctly I think I can multiply these tensors with . and Transpose.

As @yarchik has pointd out, I have to add that my tensors have unit length

You can write it directly as you see it

xxv[i_,j_,k_]:= 1/6( x[i]x[j]v[k]+x[i]x[k]v[j]
 +x[j]x[i]v[k]+x[k]x[j]v[i]
 +x[j]x[k]v[i]+x[k]x[i]v[j] )
 -1/5( KroneckerDelta[i,j]Sum[x[l]x[l],l,3]v[k]
 +KroneckerDelta[i,k]Sum[x[l]v[l],l,3]x[j]
 +KroneckerDelta[j,k]Sum[x[l]v[l],l,3]x[i] )

FullSimplify[ Sum[xxv[i,j,k] xxv[i,j,k],i,3,j,3,k,3],
 Assumptions->Sum[x[i]^2,i,3]==1
 &&Sum[x[i]v[i],i,3]==xv
 &&Sum[v[i]v[i],i,3]==vv]

Out[1]= 2/25 (4 vv + xv^2)

where I assumed that your vector x is normalized

This is how I'd do it; maybe it's useful for you.

Define $vecX$ and $vecV$ as vectors:

X = Array[x, 3];
V = Array[v, 3];

useful $3times3times3$ tensors for assembling:

a = Outer[Times, X, X, V];
b = (X.X) Outer[Times, IdentityMatrix[3], V];
c = (X.V) Outer[Times, IdentityMatrix[3], X];

assemble $XXV$:

XXV = (a + Transpose[a, 3, 1, 2] + Transpose[a, 2, 3, 1])/3 -
 (b + Transpose[c, 3, 1, 2] + Transpose[c, 2, 3, 1])/5;

check a formula:

Total[XXV*XXV, 3] == 2/25 (X.X) ((X.V)^2 + 4 (X.X) (V.V)) // FullSimplify
(* True *)

You can cast this as a symbolic tensor question, and make use of my TensorSimplify package. Install the paclet with:

PacletInstall[
 "TensorSimplify", 
 "Site" -> "http://raw.githubusercontent.com/carlwoll/TensorSimplify/master"
]

Once installed, load the package with:

<<TensorSimplify`

Now, define your tensor using TensorProduct:

XXV = 1/3 (TensorProduct[X,X,V] + TensorProduct[X,V,X] + TensorProduct[V,X,X]) - 
 1/5 (X.X TensorProduct[Inactive[IdentityMatrix][3], V] + 
 X.V TensorTranspose[TensorProduct[Inactive[IdentityMatrix][3],X],1,3,2] + 
 X.V TensorProduct[X,Inactive[IdentityMatrix][3]]
 );

Note the use of Inactive[IdentityMatrix][3] instead of IdentityMatrix[3]. Then:

TensorSimplify[
 TensorContract[TensorProduct[XXV, XXV], 1, 4, 2, 5, 3, 6],
 Assumptions -> (X|V) ∈ Vectors[3]
]

2/25 (V.X)^2 X.X + 8/25 V.V (X.X)^2

Using X.X == 1 reproduces your result.