$${{\frac{\Gamma \left( c+1 \right) \Gamma \left( a+b+c+d \right) G^{1, 3}{3, 3}_{3, 3}\left(-1\, {|}^{0, \Big\vert}^{0, 1-a-c, 1-c-d}{0, 1-c-d}_{0, 1-a-b-c-d, -c}\right) }{\Gamma \left( a+c \right) \Gamma \left( c+d \right) }}}$$ where $G$ is the Meijer G function. But I don't know if you'd call that more "closed-form" than the original hypergeometric.
$${{\frac{\Gamma \left( c+1 \right) \Gamma \left( a+b+c+d \right) G^{1, 3}{3, 3}\left(-1\, {|}^{0, 1-a-c, 1-c-d}{0, 1-a-b-c-d, -c}\right) }{\Gamma \left( a+c \right) \Gamma \left( c+d \right) }}}$$ where $G$ is the Meijer G function. But I don't know if you'd call that more "closed-form" than the original hypergeometric.