>> syms a b x
>> syms n positive
>> int(exp(-a*x)*(1 + x*1i/b)^(-n), x,0,inf)
ans =
piecewise(angle(1i/b) < pi & (angle(a) in Dom::Interval(-pi/2, pi/2) | angle(a) in {-pi/2, pi/2}) & a ~= 0, (b*exp(-(a*b*1i)/2)*(a*b*1i)^(n/2)*whittakerM(n/2, 1/2 - n/2, a*b*1i))/(a*b - a*b*n) + (pi*exp(-a*b*1i)*(-a*b*1i)^n)/(a*sin(pi*n)*gamma(n)), angle(1i/b) == pi | ~angle(a) in Dom::Interval(-pi/2, pi/2) & ~angle(a) in {-pi/2, pi/2} | a == 0, int(exp(-x*a)/((x*1i)/b + 1)^n, x, 0, Inf), angle(1i/b) < pi & (angle(a) in Dom::Interval(-pi/2, pi/2) | angle(a) in {-pi/2, pi/2}) & a ~= 0 & n == 1/2, (b*pi^(1/2)*exp(-a*b*1i)*(erf((-a*b*1i)^(1/2)) - 1)*1i)/(-a*b*1i)^(1/2))
You can get an unconditional result if you are able to put constraints on a and b, such as
syms a b n positive
syms x
Best Answer