An integral hidden in Gradshteyn and Ryzhik

We provide a closed-form expression for the integral N0,4(a;m):=∫0∞dz(z4+2az2+1)m+1 where m ∈ N and a ∈ (−1,∞): N0,4(a;m)=π2m+3/2(a+1)m+1/2Pmm+1/2,-m-1/2)(a)=π23m+32(a+1)m+12x∑k=0m2k2m−2km−km+km (a + 1)k. Here Pm(m+12,−m−12)(a) is the Jacobi polynomial Pm(α,β)(a) with parameters α=m+12 and β=−(m+12).