I try to make the following formula as simple as possible.

`(\[Theta]*Subscript[\[Alpha], 1]*(2 + (1/2)*(\[Theta]*Subscript[\[Alpha], 1] + Sqrt[8*\[Theta]*Subscript[\[Alpha], 1] + \[Theta]^2*Subscript[\[Alpha], 1]^2])))/ ((1/2)*(\[Theta]*Subscript[\[Alpha], 1] + Sqrt[8*\[Theta]*Subscript[\[Alpha], 1] + \[Theta]^2*Subscript[\[Alpha], 1]^2]) + \[Theta]*Subscript[\[Alpha], 1]* (2 + (1/2)*(\[Theta]*Subscript[\[Alpha], 1] + Sqrt[8*\[Theta]*Subscript[\[Alpha], 1] + \[Theta]^2*Subscript[\[Alpha], 1]^2]))) `

So. I use `Simplify`

and get the result.

`(\[Theta]*Subscript[\[Alpha], 1]*(4 + \[Theta]*Subscript[\[Alpha], 1] + Sqrt[\[Theta]*Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])]))/ (\[Theta]^2*Subscript[\[Alpha], 1]^2 + Sqrt[\[Theta]*Subscript[\[Alpha], 1]* (8 + \[Theta]*Subscript[\[Alpha], 1])] + \[Theta]*Subscript[\[Alpha], 1]* (5 + Sqrt[\[Theta]*Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])])) `

This is the thing I can calculate by hands as well. However, if I use`FullSimplify`

then it gives

`(4*\[Theta]*Subscript[\[Alpha], 1])/(3*\[Theta]*Subscript[\[Alpha], 1] + Sqrt[\[Theta]*Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])]) `

which is more simple. However, I have a fundamental suspicion about the result from `FullSimplify`

. Thus, I check these two are indentical.

`sol3 = (\[Theta]* Subscript[\[Alpha], 1]*(4 + \[Theta]*Subscript[\[Alpha], 1] + Sqrt[\[Theta]* Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])]))/ (\[Theta]^2*Subscript[\[Alpha], 1]^2 + Sqrt[\[Theta]*Subscript[\[Alpha], 1]* (8 + \[Theta]*Subscript[\[Alpha], 1])] + \[Theta]* Subscript[\[Alpha], 1]* (5 + Sqrt[\[Theta]* Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])])) `

and

`sol4 = (4*\[Theta]*Subscript[\[Alpha], 1])/(3*\[Theta]*Subscript[\[Alpha], 1] + Sqrt[\[Theta]*Subscript[\[Alpha], 1]*(8 + \[Theta]*Subscript[\[Alpha], 1])]) {sol3, sol4} // RootReduce SameQ @@ (Sort /@ %) `

which gives `False`

. I don’t know how could we get `FullSimplify`

result from the `Simplify`

result.