Mathematica finds also a solution for c = 0
In[84]:= Integrate[
E^(-((a^2 b c^2)/(a^2 + b)))/(a^2 + b)^2 /. c -> 0, a]
Out[84]= a/(2 b (a^2 + b)) + ArcTan[a/Sqrt[b]]/(2 b^(3/2))
In[86]:= Integrate[
Series[E^(-((a^2 b c^2)/(a^2 + b)))/(a^2 + b)^2, {c, 0, 12}],
a] // FullSimplify
Out[86]= SeriesData[c, 0, {
Rational[1, 2] b^Rational[-3, 2] (
a b^Rational[1, 2]/(a^2 + b) + ArcTan[a b^Rational[-1, 2]]), 0,
Rational[1,
8] ((a^2 + b)^(-2) (-a^3 + a b) - b^Rational[-1, 2] ArcTan[
a b^Rational[-1, 2]]), 0,
Rational[1, 96] a (a^2 - 3 b) b (a^2 + b)^(-3) (
3 a^2 + b) + Rational[1, 32] b^Rational[1, 2] ArcTan[
a b^Rational[-1, 2]], 0,
Rational[1, 2304]
a b^2 (a^2 + b)^(-4) ((-15)
a^6 + 73 a^4 b + 55 a^2 b^2 + 15 b^3) + Rational[-5, 768]
b^Rational[3, 2] ArcTan[a b^Rational[-1, 2]], 0,
Rational[-1, 92160]
a b^3 (a^2 + b)^(-5) ((-105)
a^8 + 790 a^6 b + 896 a^4 b^2 + 490 a^2 b^3 + 105 b^4)\
+ Rational[7, 6144] b^Rational[5, 2] ArcTan[a b^Rational[-1, 2]], 0,
Rational[1, 1843200] b^Rational[7, 2] (
a b^Rational[1, 2] (
a^2 + b)^(-6) ((-315)
a^10 + 3335 a^8 b + 5058 a^6 b^2 + 4158 a^4 b^3 + 1785 a^2 b^4\
+ 315 b^5) - 315 ArcTan[a b^Rational[-1, 2]]), 0,
Rational[1, 154828800]
b^Rational[
9, 2] (-a b^Rational[1, 2] (
a^2 + b)^(-7) ((-3465)
a^12 + 48580 a^10 b + 92323 a^8 b^2 + 101376 a^6 b^3 + 65373 a\
^4 b^4 + 23100 a^2 b^5 + 3465 b^6) + 3465 ArcTan[
a b^Rational[-1, 2]])}, 0, 13, 1]
