While reading a number theory paper I encountered the identity

$$ \int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$

apparently true for all $z \in \bf{C}$ for which the integral on the left converges absolutely. The author offers neither a justification nor a citation, so apparently this is 'well known'.

I thought of two ideas for how to prove it: (1) Use integration by parts to prove that both sides satisfy the same functional equation, and then attempt to adapt the proof of the Bohr-Mollerup theorem to prove that this functional equation characterizes the function; (2) use integration by parts to analytically continue the left-side function in $z$, determine all the poles and their residues, and attempt to use Weierstrass factorization to prove that this characterizes the function.

Both approaches looked rather involved and I didn't attempt to work out the details of either. Is this, indeed, 'well known'? Is there a suitable reference which proves this, along with other identities of this nature? And to what extent does the identity generalize?