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Definite integral of Bessel function of the first kind times $x^{3/2}$

The integral requires $\nu>-5/2$ for convergence, and then becomes a hypergeometric function: $$\int_0^a x^{3/2} J_\nu (bx) dx=\frac{2^{1-\nu} a^{\nu+\frac{5}{2}} b^{\nu}}{(2 \nu+5) \Gamma (\nu+1)}\...
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