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Any idea how to solve analytically this integral

Integrate[(a Erf[a Sqrt[b/(a^2 + b)] c])/(a^2 + b)^(3/2), a]

I tried substitution u=a^2 + b, but it didn't work. I don't need numerical integration or to solve via the few terms of the Taylor expansion. This seems to be very difficult to solve, but maybe somebody has got an idea.

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The answers to this question might shed some light on the issue: Numerical underflow for a scaled error function. – Artes Feb 3 '14 at 0:16
up vote 3 down vote accepted

Some integrals cannot be found in terms of a finite set of functions. I'm not an expert on what the limits of Mathematica are, but the following suggests to me that this integral is beyond them.

The reason comes down, by a sequence of substitutions, to the fact that this integral returns unaltered:

-Integrate[Erf[Sqrt[1 - w^2]], w]
  -Integrate[Erf[Sqrt[1 - w^2]], w]

The sequence of substitutions

integrand = (a Erf[a Sqrt[b/(a^2 + b)] c])/(a^2 + b)^(3/2) Dt[a];
sub = Solve[u == a Sqrt[b/(a^2 + b)], a]
  {{a -> -((Sqrt[b] u)/Sqrt[b - u^2])}, {a -> (Sqrt[b] u)/Sqrt[b - u^2]}}

int2 = Simplify[
  integrand /. Last@sub,
  Dt[b] == 0 && b > 0 && c > 0 && u > 0]
   (u Sqrt[1/(b - u^2)] Dt[u] Erf[c u])/b

sub3 = Solve[w^2 == b - u^2, u]
  {{u -> -Sqrt[b - w^2]}, {u -> Sqrt[b - w^2]}}

int3 = Simplify[
  int2 /. Last@sub3,
  Dt[b] == 0 && b > 0 && c > 0 && w > 0]
  -((Dt[w] Erf[c Sqrt[b - w^2]])/b)

Integrate[int3 /. Dt[w] -> 1, w]
  -(Integrate[Erf[c Sqrt[b - w^2]], w]/b)

This last integral is nearly the same as the first one. Somehow I tend to believe that if the integral could be done, Mathematica would do it.

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