I assume that the eta after the exponential in the first term is not in the exponential but is simply multiplying it. If you are intersted in only the real part of your J7 expression then that's simply the first term as you've written it for real values of eta.

So this is that integral:

In[20]:= (2/\[Pi])^(1/2)

Integrate[\[Eta] x^7 Exp[-x^2/2]/(\[Eta]^2 + x^10), {x,

0, \[Infinity]}, Assumptions -> {\[Eta] > 0}]

Out[20]= (\[Eta]^(3/5)

MeijerG[{{1/5}, {}}, {{0, 1/5, 1/5, 2/5, 3/5, 4/5}, {}}, \[Eta]^2/

100000])/(4 Sqrt[10] \[Pi]^(5/2))

The result in closeed form. Now all you need to do is substitute for eta and plot the result. However you do not say what the value of omega is. The expression evaluated for your specified (omega dependent) value for lambda

In[26]:= (\[Eta]^(3/5)

MeijerG[{{1/5}, {}}, {{0, 1/5, 1/5, 2/5, 3/5, 4/5}, {}}, \[Eta]^2/

100000])/(

4 Sqrt[10] \[Pi]^(5/2)) /. {\[Eta] -> 0.28 10^5 \[Omega] \[Lambda]}

Out[26]= 2.10552 (\[Lambda] \[Omega])^(3/5)

MeijerG[{{1/5}, {}}, {{0, 1/5, 1/5, 2/5, 3/5, 4/5}, {}},

7840. \[Lambda]^2 \[Omega]^2]