Why can't wolfram alpha solve this simple quintic?

Question

So I found out that some transformations break wolfram alpha's ability to solve polynomials. The simplest case I could find is the polynomial $$2x^{5}+5x^{4}+5x^{2}+1=0$$ for which the solution is $$x=\frac{1}{\sqrt[5]{\sqrt{2}-1}-\sqrt[5]{\sqrt{2}+1}}=\frac{\sqrt[5]{12\sqrt{2}-17}-\sqrt[5]{12\sqrt{2}+17}-\sqrt[5]{2\sqrt{2}+3}+\sqrt[5]{2\sqrt{2}-3}-1}{2}$$ Which derives simply from the solvable Moivre quintic $$x^{5}+5ax^{3}+5a^{2}x+2b=0,\ x=\sqrt[5]{\sqrt{a^{5}+b^{2}}-b}-\sqrt[5]{\sqrt{a^{5}+b^{2}}+b}$$ which wolfram alpha can solve. But for some reason, when I try this polynomial, wolfram alpha can only approximate it. Weirder, wolfram alpha is entirely capable of solving other quintics with a sum of $4$ radicals and a rational number, like $$x^{5}+5x^{4}+10x^{3}+10x^{2}+8=0,\ x=\frac{\sqrt[5]{750\sqrt{25+5\sqrt{5}}+375\sqrt{25-5\sqrt{5}}-1250\sqrt{5}-3125}+\sqrt[5]{375\sqrt{25+5\sqrt{5}}-750\sqrt{25-5\sqrt{5}}+1250\sqrt{5}-3125}-\sqrt[5]{750\sqrt{25+5\sqrt{5}}+375\sqrt{25-5\sqrt{5}}+1250\sqrt{5}+3125}-\sqrt[5]{375\sqrt{25+5\sqrt{5}}-750\sqrt{25-5\sqrt{5}}-1250\sqrt{5}+3125}-5}{5}$$ and that example is considerably more complicated than this $1$. A few more sophisticated examples exist, like $ax^{5}+bx^{2}+c=0$, which is solved with hypergeometric functions. Wolfram alpha does use hypergeometric functions to solve other polynomials, like $ax^{5}+bx+c=0$, so what gives?