Answer: $\frac{49}{16} \approx 3.06$.

Solution. The equation is equivalent to the following: $(x+1)^{3 / 2}=\frac{3}{2}(x+1)^{5 / 4}$, or $(x+1)\left((x+1)^{1 / 4}-\frac{3}{2}\right)=$ 0 , from which we find the roots: $x=-1, x=\frac{81}{16}-1=\frac{65}{16}$. Their sum is $\frac{49}{16} \approx 3.06$.
6.1.1. (12 points) Solve the equation

$$
5^{\sqrt{x^{3}+3 x^{2}+3 x+1}}=\sqrt{\left(5 \sqrt[4]{(x+1)^{5}}\right)^{3}}
$$

In the answer, write the root if there is only one, or the sum of the roots if there are several.