I have the following cubic equation with $ \beta \in [0,1]$ and $ \delta\in [0,1]$ are 2 parameters. Is it possible to use software to get the explicit expression of the solution.
\begin{equation*} U^{3}\left( -\left( \frac{\delta^{2}}{2} + \beta\right)+\delta-\frac{1}{2} \right)+U^{2}\left(\frac{3}{2}-\delta-\left( \frac{\delta^{2}}{2} + \beta\right)\right)+U\left(\left( \frac{\delta^{2}}{2}+\beta -2 \delta\right)\right)+\left( \frac{\delta^{2}}{2} + \beta\right)=0 \end{equation*}
I want to solve this equation usin Maple or Mathematica, but I was not able to do it.
You can find formula for cubic equation in Wikipedia: https://en.wikipedia.org/wiki/Cubic_equation
I will shortly describe the process how to solve such equation. This can be done in steps to minimize the use of long complicated expressions.
Step 1: write $aU^3 + bU^2 + cU + d = 0$.
Step 2: reduce to the case $T^3 + pT + q = 0$ (as explained in Wikipedia, divide the equation by $a \ne 0$ and then subsitute $U = T - \frac{b}{3a}$).
Step 3: use Cardano's formula. Taking the real cubic roots you get $$ T_{\mathbb{R}}=u+v=\sqrt[3]{-{q\over 2}+ \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}} +\sqrt[3]{-{q\over 2}- \sqrt{{q^{2}\over 4}+{p^{3}\over 27}}} $$
Step 4: to find the rest of the roots divide $T^3 + pT + q$ by $T-T_{mathbb{R}}$ and then solve the quadratic that you get.
In all the steps you intermediate parameters-variables, e.g. $$ A = -\left( \frac{\delta^2}{2} + \beta \right) + \delta - \frac{1}{2} $$ etc.