Use Maple to solve the equation x^5+x^4−12x^3−21x^2+x+5=0 y^5+y^4−16y^3+5y^2+21y−9=0 y^5+y^4−24y^3−17y^2+41y−13=0 y^5+y^4−28y^3+37y^2+25y+1=0 10x^6-75x^3-190x+21=0 All the previous quintic equations have radical solutions The last one, I don't know if there is. If there is, we need radical solution No, I hope it can be solved with a special function, like this z^5+z^4-e^6=0 z=exp(6/5)*hypergeom([-1/5,1/20,3/10,11/20],[1/5,2/5,3/5],256/3125*exp(-6)) +2/25*exp(-6/5)* hypergeom([1/5,9/20,7/10,19/20],[3/5,4/5,7/5],2/3125*exp(-6)) -4/125*exp(-12/5)* hypergeom([2/5,13/20,9/10,23/20],[4/5,6/5,8/5],256/3125*exp(-6)) +7/625*exp(-18/5)* hypergeom([3/5,17/20,11/10,27/20],[6/5,7/5,9/5],256/3125*exp(-6)) -1/5

Use Maple to solve the equation x^5+x^4−12x^3−21x^2+x+5=0 y^5+y^4−16y^3+5y^2+21y−9=0 y^5+y^4−24y^3−17y^2+41y−13=0 y^5+y^4−28y^3+37y^2+25y+1=0 10x^6-75x^3-190x+21=0 All the previous quintic equations have radical solutions The last one, I don't know if there is. If there is, we need radical solution No, I hope it can be solved with a special function, like this z^5+z^4-e^6=0 z=exp(6/5)*hypergeom([-1/5,1/20,3/10,11/20],[1/5,2/5,3/5],256/3125*exp(-6)) +2/25*exp(-6/5)* hypergeom([1/5,9/20,7/10,19/20],[3/5,4/5,7/5],2/3125*exp(-6)) -4/125*exp(-12/5)* hypergeom([2/5,13/20,9/10,23/20],[4/5,6/5,8/5],256/3125*exp(-6)) +7/625*exp(-18/5)* hypergeom([3/5,17/20,11/10,27/20],[6/5,7/5,9/5],256/3125*exp(-6)) -1/5

You can do it yourself with maple, like the first equation
allvalues({solve(x^5+x^4-12*x^3-21*x^2+x+5=0)});
The first four are radical solutions. The answer is very long
The last one can't get the radical solution, and I don't know how to express it with special function. Sorry