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To get your Integral(s), You can use Wolfram|Alpha, see Wolfram|Alpha Integration, see also How can I get all the results from a Wolfram | Alpha query?, You ca also use http://www.wolframalpha.com/

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?? MarcumQ Then click on the >> to go to the reference page. Then open the Details panel to see: $Q_m(a,b) = \int\limits_b^\infty x \left( {x \over a}\right)^{m-1} I_{m-1}(a x) Exp[- {1 \over 2} (a^2 + x^2)]\ dx$

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we need to force Integrate function to evaluate w.r.t x ClearAll[f, g, x] g[x_] := 3 x^2 + 2 x f[x_] := Evaluate[20 x + Integrate[g[x], x]] f[2] (* 52 *)

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In response to your second question, "Is there a way to tell MMa to convert (for example) all the Hypergeometric functions into whatever Bessel function form it can?", execute $Post = FunctionExpand[#] & at the beginning of a Notebook to cause FunctionExpand to be applied to the output of each Cell after it is executed. However, be aware that$Post = ...

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