Integral with 4 Bessel functions and an exponential

Let's consider the second integral, which can be written in the following form: $$ I(p, q, i, j, k, l; a, b, c, d) := int_0^infty dt, exp(-p t^2) t^q j_i(a t) j_j(b t) j_k(c t) j_l(d t) $$ where in your case, $i = j = k = 1$, $l = 0$, and $q = -1$.

These kinds of integrals (as well their generalization to a product of arbitrarily many spherical Bessel functions) are discussed in Fabrikant - Elementary exact evaluation of infinite integrals of the product of several spherical Bessel functions, power and exponential, where the main idea is to use the following identity: begin{align} I(p, q, i, j, k, l; a, b, c, d) &= (-1)^{i+j+k+l} a^i b^j c^k d^l frac{partial^i}{(a partial a)^i} frac{partial^j}{(b partial b)^j} frac{partial^k}{(c partial c)^k} frac{partial^l}{(d partial d)^l} biggl[\ &int_0^infty dt exp(-p t^2) frac{ j_0(a t) j_0(b t) j_0(c t) j_0(d t) } { t^{i + j + k + l - q} } biggr]. end{align}

The key point is to now expand the zeroth order spherical Bessel functions into trigonometric functions, and converting the products of the trigonometric functions into sums: begin{align} sin(ax) sin(bx) sin(cx) sin(dx) =& frac{1}{8} biggl{ cos[(a + b + c + d)x] + cos[(a + b - c - d)x] + cos[(a - b + c - d)x]\ &+ cos[(a - b - c + d)x] - cos[(-a + b + c + d)x] - cos[(a - b + c + d)x]\ &- cos[(a + b - c + d)x] - cos[(a + b + c - d)x] biggr} end{align} followed by the use of the following integral, which is not considered in the reference above, but can be found in Gradshteyn and Ryzhik, 7th ed., formula 3.953.8: $$ mathcal{I}(p, s; n) := int_0^infty dt, t^n exp(-p t^2) cos(s t) = frac{1}{2} p^{frac{-(n + 1)}{2}} , e^{-s^2 / 4 p} Gamma left(frac{1}{2} + frac{n}{2}right) , _1F_1left(-frac{n}{2}; frac{1}{2}; frac{s^2}{4 p}right). $$

Note that the formal requirement is that $operatorname{Re}(n) > -1$, but the above result can be understood as an analytic continuation for general values $p, s, n$.

Additionally, it can happen that one of the "angles" above is zero, in which case we have the integral: $$ mathcal{I}(p, 0; n) := int_0^infty dt, t^n exp(-p t^2) = frac{1}{2} p^{-frac{n}{2}-frac{1}{2}} Gamma left(frac{n+1}{2}right) $$ with the same condition on $n$ as above.

The result in your specific case is then: begin{align} I(p, -1, 1, 1, 1, 0; a, b, c, d) &= - a b c frac{partial}{(a partial a)} frac{partial}{(b partial b)} frac{partial}{(c partial c)} int_0^infty dt exp(-p t^2) frac{ j_0(a t) j_0(b t) j_0(c t) j_0(d t) } { t^4 }\ &= - a b c frac{partial}{(a partial a)} frac{partial}{(b partial b)} frac{partial}{(c partial c)} int_0^infty dt exp(-p t^2) frac{ sin(a t) sin(b t) sin(c t) sin(d t) } { a, b, c, d, t^8 }\ &= - frac{partial}{partial a} frac{partial}{partial b} frac{partial}{partial c} bigg[ frac{1}{a, b, c, d} int_0^infty dt exp(-p t^2) frac{ 1 } { t^8 } frac{1}{8} bigg{ \ &cos[(a + b + c + d)t] + cos[(a + b - c - d)t] + cos[(a - b + c - d)t] \ &+ cos[(a - b - c + d)t] - cos[(-a + b + c + d)t] - cos[(a - b + c + d)t] \ &- cos[(a + b - c + d)t] - cos[(a + b + c - d)t] bigg} bigg] \ &= - frac{1}{8} frac{partial}{partial a} frac{partial}{partial b} frac{partial}{partial c} biggl{frac{1}{a, b, c, d} \ &mathcal{I}(p, a + b + c + d; -8) + mathcal{I}(p, a + b - c - d; -8) + mathcal{I}(p, a - b + c - d; -8) \ &+ mathcal{I}(p, a - b - c + d; -8) - mathcal{I}(p, -a + b + c + d; -8) - mathcal{I}(p, a - b + c + d; -8) \ &- mathcal{I}(p, a + b - c + d; -8) - mathcal{I}(p, a + b + c - d; -8) biggr}. end{align}

The explicit result is fairly cumbersome to fully write out; below is an example Mathematica code which can be used as a starting point to generate the full solution (when $a pm b pm c pm d neq 0$) and compare it with the numerical result:

numeric[p_, a_, b_, c_, d_] := NIntegrate[
  Exp[-p t^2] SphericalBesselJ[1, a t] SphericalBesselJ[1, 
    b t] SphericalBesselJ[1, c t] SphericalBesselJ[0, d t]/t,
  {t, 0, Infinity}
  ];
integral[p_, s_, n_] := 
  1/2 p^(-(n + 1)/2) Exp[-s^2/(4 p)] Gamma[
    1/2 + n/2] Hypergeometric1F1[-n/2, 1/2, s^2/(4 p)];
result = -1/8 Table[
    Series[
       expression,
       {epsilon, 0, 0}
       ] // Normal // D[#/(a b c d), a, b, c] &,
    {
     expression,
     {
      integral[p, a + b + c + d, -8 + epsilon],
      integral[p, a + b - c - d, -8 + epsilon],
      integral[p, a - b + c - d, -8 + epsilon],
      integral[p, a - b - c + d, -8 + epsilon],
      -integral[p, -a + b + c + d, -8 + epsilon],
      -integral[p, a - b + c + d, -8 + epsilon],
      -integral[p, a + b - c + d, -8 + epsilon],
      -integral[p, a + b + c - d, -8 + epsilon]
      }
     }
    ] // Total;

No idea if the solution which the code above generates can be simplified though.