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How can I get rid of the complex numbers in this result? The actual result is obviously real, but Mathematica gives me this inconvenient complex form.

Integrate[Exp[a*(x^3)], {x, 0, b}, Assumptions -> {a > 0, b > 0}]

((1 - I Sqrt[3]) (Gamma[1/3] - Gamma[1/3, -a b^3]))/(6 a^(1/3))

And I've already seen this related question, but the methods suggested for that problem didn't work for this. ComplexExpand[] just makes it into an absolute mess, and Simply puts it right back into complex form.

How to get rid of all complex numbers and functions?

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Here is dirty trick suitable for that particular case:

rez = Integrate[Exp[5*(x^3)], {x, 0, 7}, 
   Assumptions -> {a > 0, b > 0}] // ToRadicals

rezComplex = 
 Integrate[Exp[a*(x^3)], {x, 0, b}, Assumptions -> {a > 0, b > 0}]

realAnswer = (rez // ToRadicals) /. {5 -> a, -1715 -> -a*b^3}

FullSimplify[(realAnswer - rezComplex), Assumptions -> {a > 0, b > 0}]
(* 0*)
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Try the following trick:

realPartRule = Complex[re_, im_] :> Complex[re, 0];
realPart[exp__] := exp /. realPartRule;

Applying this trick to your result we obtain:

realPart[Integrate[Exp[a*(x^3)], {x, 0, b}, 
Assumptions -> {a > 0, b > 0}]]
(*(Gamma[1/3] - Gamma[1/3, -a b^3])/(6 a^(1/3))*)
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