$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\delta (6 c+5 \text{R2}) \delta (30 a+24 \text{R1}-5 \text{R2}) \exp \left(-\frac{24 \text{R1}^2-48 \text{R1} \text{S1}+25 \text{R2}^2-50 \text{R2} \text{S2}+24 \text{S1}^2+25 \text{S2}^2}{30 b^2}\right)d\text{R2}d\text{R1}$
How to calculate this integral? This integral cannot be calculated in Mathematica (MMA).
I believe the integral can be rearranged as
$$I=\int\limits_{-\infty}^{\infty} \delta(6 c+5 R_2) \left(\int\limits_{-\infty}^{\infty} \delta(30 a+24 R_1-5 R_2) \exp\left(-\frac{24 R_1^2-48 R_1 S_1+25 R_2^2-50 R_2 S_2+24 S_1^2+25 S_2^2}{30 b^2}\right) \, dR_1\right) \, dR_2$$
which can then be evaluated using the composition with a function property of $\delta(g(x))$.
Mathematica actually gives the result
$$I=\frac{1}{120} \exp \left(-\frac{75 a^2+30 a (c+4 S_1)+75 c^2+24 c (S_1+5 S_2)+48 S_1^2+50 S_2^2}{60 b^2}\right)$$
for this integral assuming $a\in\mathbb{R}\land c\in\mathbb{R}$ which agrees with evaluation using the composition property.