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The integral  \[    I v α  , β  ( x ) = ∫  0 ∞  t 2 α  −  1 ( 1 + t 2 ) 1 −  α  −  β  J v ( x 1 + t 2 ) d t I_v^{\alpha ,\beta }(x) = \int _0^\infty {{t^{2\alpha - 1}}{{(1 + {t^2})}^{1 - \alpha - \beta }}{J_v}(x} \sqrt {1 + {t^2})} \;dt     \]  is expressed in terms of Bessel and related functions for various values of the parameters by summing the hypergeometric series representation given by Schmidt.

A note on the integral ∫^{∞}₀ 𝑡^{2𝛼-1}(1+𝑡²) ^{1-𝛼-𝛽}𝐽_{𝜈}(𝑥√(1+𝑡²))𝑑𝑡