= Solution
Let $M_{ijkl}=\int_Vx_ix_jx_kx_l\,d^3x$. It is an <isotropic tensor> and is invariant under every permutation of its four indices. Thus the three coefficients in the general rank-four isotropic form must coincide:
$$
M_{ijkl}=C(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}).
$$
A double <tensor contraction> gives
$$
15C=M_{iijj}=\int_Vr^4\,d^3x
=4\pi\int_{R_1}^{R_2}r^6\,dr
=\frac{4\pi}{7}(R_2^7-R_1^7).
$$
Here the three contracted <Kronecker delta> products contribute $9$, $3$ and $3$. Contracting instead with the four unit <vectors> yields
$$
\boxed{\begin{aligned}
&\int_V\prod_{a=1}^4(\mathbf x\cdot\mathbf n_a)\,d^3x\\
&\quad=\frac{4\pi}{105}(R_2^7-R_1^7)
\big[(\mathbf n_1\cdot\mathbf n_2)(\mathbf n_3\cdot\mathbf n_4)
+(\mathbf n_1\cdot\mathbf n_3)(\mathbf n_2\cdot\mathbf n_4)
+(\mathbf n_1\cdot\mathbf n_4)(\mathbf n_2\cdot\mathbf n_3)\big].
\end{aligned}}
$$
This agrees with subtracting two instances of the <isotropic fourth moment of a ball>.
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