5. Given a tetrahedron $ABCD$ such that $AB=\sqrt{3}, AD=$ $BC=\sqrt{10}, AC=CD=BD=\sqrt{7}$. The volume of the tetrahedron is $\qquad$.
5. $\sqrt{2}$.

Let $A(0,0,0), B(\sqrt{3}, 0,0), C(0, \sqrt{7}, 0)$, $D(x, y, z)$. Then
$$
\left\{\begin{array} { l } 
{ x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1 0 , } \\
{ ( x - \sqrt { 3 } ) ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 7 , } \\
{ x ^ { 2 } + ( y - \sqrt { 7 } ) ^ { 2 } + z ^ { 2 } = 7 }
\end{array} \Rightarrow \left\{\begin{array}{l}
x=\sqrt{3}, \\
y=\frac{5}{\sqrt{7}}, \\
z=\frac{2 \sqrt{6}}{\sqrt{7}} .
\end{array}\right.\right.
$$

Thus, $V=\frac{1}{3} \times \frac{1}{2} \times \sqrt{3} \times \sqrt{7} \times \frac{2 \sqrt{6}}{\sqrt{7}}=\sqrt{2}$.