Does there exist any?

Given that function $f(x)=(x^3-mx)\ln(x^2+1-m) \ (m \in \mathbb R)$ has exactly $3$ distinct roots.

Does there exist any $m \in \mathbb R$ that satisfies the above condition and $f'(x)=0$ has exactly $2$ distinct roots $x_1,x_2$ for $x \in (0,1)$ and $x_2=2x_1$ ?