Evaluate integrals with parameters

Question

I'm new to Mathematica so I apologize if the answer to my question is trivial. I need to calculate the following integral $$ \int_{-\infty}^{0}\frac{dx}{\sqrt{(x-a)(x-b)(x-c)(x-d)}} $$ with $$ 0 < d < c < b < a $$ (of course $a,b,c$ and $d$ are real)

Is it possible to obtain a results in terms of $a,b,c$ and $d$?

And if it is possible how should I do?

Answer 1

This is a kind of elliptic integral, and it can be calculated if we prescribe appropriate assumptions.

Integrate[ 1/Sqrt[(x - a) (x - b) (x - c) (x - d)], {x, -Infinity, 0}, 
           Assumptions -> 0 < d < c < b < a]

-((2(EllipticF[ArcSin[Sqrt[(a - d)/(b - d)]], ((a - c) (b - d))/((b - c) (a - d))]
   - EllipticF[ArcSin[Sqrt[(b (a - d))/(a (b - d))]], 
               ((a - c) (b - d))/((b - c) (a - d))]))/Sqrt[(b - c) (a - d)])

EllipticF is the incomplete elliptic integral of the first kind. Such an integral can be also expressed in terms of the Weierstrass elliptic function, precisely by its inverse.

Answer 2

Artes shows that the integral can be expressed in terms of the Legendre-Jacobi integrals. I will now present a solution that uses the Carlson elliptic integrals, for which I have written a package implementing them.

In particular, our working formulae are a combination of formulae 19.29.4 and 19.29.6 in the DLMF:

<<Carlson` (* load package after installing *)

2 (CarlsonRF @@ (Total[(Apply[Times, Sqrt[{a, b, c, d}[[#]]]] & /@
                       {#, Complement[Range[4], #]}) & /@ Subsets[Range[3], {2}], {2}]^2))
   2 CarlsonRF[(Sqrt[b] Sqrt[c] + Sqrt[a] Sqrt[d])^2,
               (Sqrt[a] Sqrt[c] + Sqrt[b] Sqrt[d])^2,
               (Sqrt[a] Sqrt[b] + Sqrt[c] Sqrt[d])^2]

and the result can be seen to have permutation symmetry in a,b,c,d.

Compare:

With[{a = 2, b = 3, c = 4, d = 5},
     {N[2 CarlsonRF[(Sqrt[b] Sqrt[c] + Sqrt[a] Sqrt[d])^2,
                    (Sqrt[a] Sqrt[c] + Sqrt[b] Sqrt[d])^2,
                    (Sqrt[a] Sqrt[b] + Sqrt[c] Sqrt[d])^2], 25],
      NIntegrate[1/Sqrt[(a - t) (b - t) (c - t) (d - t)], {t, -∞, 0}, 
                 Method -> "DoubleExponential", WorkingPrecision -> 25]}]
   {0.2963150890686989232562657, 0.2963150890686989232562657}