how do you integrate 1/ ( 6+x+ x^2 )^(1/2)

my working:

let 6+x+ x^2 = (x+2)^2 (z^2)

then dx = -10z / ( z^2 +1) ^(2) dz

substituting that into the question, i got

integrate -2 / (z^2+1) and hence the answer was -2 arctan ( ((3-x)/ (x+2))^(1/2))

however the answer should have been arcsin ( (2x-1)/5)).

is there something wrong with my substitution?