An integration-by-parts yielded the result \begin{equation} -\frac{\Gamma \left(\frac{d}{2}+1\right)^2 \Gamma (d) \left(\Gamma (d+1)^2 \, _3\tilde{F}_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;\varepsilon ^2\right)-2 \, _2F_1\left(-\frac{d}{2},\frac{d}{2};\frac{d+2}{2};\varepsilon ^2\right)\right)}{2 \Gamma (d+1)^2}. \end{equation}
Added together with a set of auxiliary results, this leads to a formula for the "Lovas-Andai function" (https://arxiv.org/abs/1610.01410) \begin{equation} \tilde{\chi}_d (\varepsilon )=\frac{\varepsilon ^d \Gamma (d+1)^3 \, _3\tilde{F}_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;\varepsilon ^2\right)}{\Gamma \left(\frac{d}{2}+1\right)^2}. \end{equation}\begin{equation} \tilde{\chi}_d (\varepsilon )=\frac{\varepsilon ^d \Gamma (d+1)^3 \, _3\tilde{F}_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;\varepsilon ^2\right)}{\Gamma \left(\frac{d}{2}+1\right)^2}, \end{equation} where the tilde is used to denote the regularized hypergeometric function.