= Solution
For the <spherical Abel deprojection>, put $A_r=r^2+b^2$ and $s=(R^2+b^2)/A_r$. Then $dR^2=A_r\,ds$ and $\sqrt{R^2-r^2}=A_r^{1/2}\sqrt{s-1}$. Consequently the integral appearing in the reconstruction of $n$ is
$$
J_n(r)=N_0b^5 A_r^{-2}I_{5/2},\qquad
I_\alpha=\int_1^\infty s^{-\alpha}(s-1)^{-1/2}\,ds.
$$
Differentiate before substituting the numerical value of $I_{5/2}$:
$$
n(r)=-\frac{J_n'(r)}{2\pi r}=\frac{2N_0b^5}{\pi A_r^3}I_{5/2}
=\boxed{\frac{8N_0b^5}{3\pi(r^2+b^2)^3}.}
$$
For the projected second <stellar velocity moment>, $N\sigma^2=N_0\sigma_0^2b^6(R^2+b^2)^{-3}$. The identical substitution in <isotropic stellar pressure deprojection> gives
$$
J_p=N_0\sigma_0^2b^6A_r^{-5/2}I_3,\qquad
p(r)=\frac{5N_0\sigma_0^2b^6}{2\pi A_r^{7/2}}I_3
=\boxed{\frac{15N_0\sigma_0^2b^6}{16(r^2+b^2)^{7/2}}.}
$$
For completeness, setting $t=1/s$ yields $I_\alpha=B(1/2,\alpha-1/2)$, where $B$ is the <Euler beta function>. The <Gamma function recurrence> gives $I_{5/2}=4/3$ and $I_3=3\pi/8$. This also checks the normalization of the <power-law spherical projection kernel>.
The radial <Jeans equation> is $p'=n\psi'$. Since $p'=-105N_0\sigma_0^2b^6r/(16A_r^{9/2})$, division by the tracer <number density> gives
$$
\psi'(r)=-\frac{315\pi}{128}\frac{\sigma_0^2br}{(r^2+b^2)^{3/2}}.
$$
Writing $A=315\pi\sigma_0^2b/128$, the total <gravitational acceleration> is therefore
$$
\boxed{\nabla\psi=-\frac{A\mathbf r}{(r^2+b^2)^{3/2}}.}
$$
To find all gravitating <mass density>, apply the <Poisson equation for Newtonian gravity>, with the same sign convention:
$$
\rho_{\rm tot}(r)=-\frac1{4\pi G r^2}\frac d{dr}(r^2\psi')
=\frac{3Ab^2}{4\pi G(r^2+b^2)^{5/2}}
=\boxed{\frac{945\sigma_0^2b^3}{512G(r^2+b^2)^{5/2}}.}
$$
Equivalently, the enclosed <mass> is $M(r)=Ar^3/[G(r^2+b^2)^{3/2}]$, and $\rho_{\rm tot}=M'/(4\pi r^2)$. The inferred total <mass> is $A/G$ and its <relative potential> is $A/\sqrt{r^2+b^2}$, a <Plummer model>. The tracer <number density> has a different radial exponent from this total <mass density>. One must not multiply $n$ by an arbitrary stellar mass and identify it with all matter: the inferred field can include unobserved <stars>, gas, and <dark matter>.
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