Is there a way to compute the following integral numerically?,

$\int_0^{\infty}\left(-\frac{2 x \Gamma \left(\frac{1}{4}\right) \, _1F_3\left(\frac{1}{4};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x^4}{256}\right)}{\sqrt{\pi }}+x^2 \, _1F_3\left(\frac{1}{2};\frac{3}{4},\frac{5}{4},\frac{3}{2};\frac{x^4}{256}\right)+\sqrt{\frac{2}{\pi }} \Gamma \left(\frac{1}{4}\right) \Gamma \left(\frac{3}{4}\right)\right)dx$ enter image description here

We know that $-\frac{2 x \Gamma \left(\frac{1}{4}\right) \, _1F_3\left(\frac{1}{4};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x^4}{256}\right)}{\sqrt{\pi }}+x^2 \, _1F_3\left(\frac{1}{2};\frac{3}{4},\frac{5}{4},\frac{3}{2};\frac{x^4}{256}\right) \rightarrow -\sqrt{\frac{2}{\pi }} \Gamma \left(\frac{1}{4}\right) \Gamma \left(\frac{3}{4}\right)$ as $x \rightarrow \infty.$

  • $\begingroup$ Mathematica could not handle to compute this integral $\endgroup$ – Dilruk Gallage Jul 20 '17 at 6:16
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    $\begingroup$ What is $g(x)$? Is it $F_3$? $\endgroup$ – MathArt Jul 20 '17 at 6:26
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    $\begingroup$ Could you post the Mathematica syntax since, being almost blind, I have serious problems to read it ? I should try with other tools. $\endgroup$ – Claude Leibovici Jul 20 '17 at 6:26
  • $\begingroup$ $g(x)$ is the integrand $\endgroup$ – Dilruk Gallage Jul 20 '17 at 6:40
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    $\begingroup$ The integral is zero. Indeed the value from 0 to 1 is almost identical to the opposite of the value from 0 to 20. The "tail" from 20 on is almost zero $\endgroup$ – Raffaele Jul 20 '17 at 10:29

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