If $\alpha$, $\beta$ and $\gamma$ are the roots of $$x^3+2x^2+3x+3=0,$$ then what is the value of $$\left(\frac{\alpha}{\alpha+1}\right)^3+\left(\frac{\beta}{\beta+1}\right)^3+\left(\frac{\gamma}{\gamma+1}\right)^3?$$
This problem has an answer of $44$ but I am unable to find it. I had my progress on paper but gave it at the contest. Then, I tried using ChatGPT to find the answer and got $-5/2$ as the answer. Please can anyone help me figure this problem?
Reference: Test of mathematics at 10+2 level, isi problem book, pg-39, problem-272.
Note: I tried to substitute the values $\alpha/(\alpha+1)$, etc., in the equation to get a pattern but this didn’t work. Also, guessing any number by hit and trial to find any one solution also did not work. I was unable to proceed further after this.