$$\int_0^z t^{\rho} J_{\mu}(at) J_{\nu}(bt) dt = \frac{(\frac 1 2 a z)^{\mu} (\frac 1 2 bz)^{\nu} z^{\rho + 1}}{\Gamma(\mu + 1)\Gamma(\nu + 1)} \times \sum_{k = 0}^{\infty} \frac{(-1)^k (\frac 1 2 a z)^{2k} {}_2 F_1(-k, -\mu - k; \nu + 1; b^2/a^2)}{k!(\mu + \nu + \rho + 2k + 1)(\mu + 1)_k}, \Re(\mu + \nu + \rho) > -1$$
I would like to solve infinite series, in simplified form.