Hi Because Mathematica dosen't know a closed form solution for define integral.

ExpectedValue[(1 + x)^2, LogNormalDistribution[\[Mu], \[Sigma]], x]
(*1 + 2 E^(\[Mu] + \[Sigma]^2/2) + E^(2 \[Mu] + 2 \[Sigma]^2)*)

is the same as:

Integrate[(1 + x)^2*
  PDF[LogNormalDistribution[\[Mu], \[Sigma]], x], {x, 0, Infinity}, 
 Assumptions -> {\[Mu] > 0, \[Sigma] > 0}]

  (*1 + 2 E^(\[Mu] + \[Sigma]^2/2) + E^(2 \[Mu] + 2 \[Sigma]^2)*)

but:

Integrate[(1 + x)^(3/10)*PDF[LogNormalDistribution[\[Mu], \[Sigma]], x], {x, 0, Infinity}, 
Assumptions -> {\[Mu] > 0, \[Sigma] > 0}]
(*Return input*)

You can solve numericaly,for example:

\[Mu] = 1; \[Sigma] = 1;
NIntegrate[(1 + x)^(3/10)*PDF[LogNormalDistribution[\[Mu], \[Sigma]], x], {x, 0, Infinity}]
(*1.56169*)

Regards M.I.