1/(2*2^(1/4)*Sqrt[Pi]*x^(3/2)) +
945*Gamma[9/4]/(256*2^(1/4)*Sqrt[Pi]*x^(11/2)*Gamma[1/4]) -
15*Gamma[5/4]/(16*2^(1/4)*Sqrt[Pi]*x^(7/2)*Gamma[1/4]) -
x^5*Gamma[13/4]/(90*Gamma[1/2]) + x^3*Gamma[9/4]/(3*Gamma[1/4]) -
2*x*Gamma[5/4]/Gamma[1/4] /. x -> 1 // N

(* 0.261485*)

F[z_] := Sqrt[z]/(z^2 + 2)^(1/4)
melsol = 1/Gamma[p] MellinTransform[F[z], z, p, GenerateConditions -> False] /. p -> (1 - p);
sol = melsol // TrigToExp // FunctionExpand;
InverseMellinTransform[sol, p, x] // FullSimplify

N[-(1/2) x HypergeometricPFQ[{5/4}, {3/2, 2}, -(x^2/2)] /. x -> 1, 20]
 (*-0.40335058864991806300*)