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Integrate[E^(a*x)*x^m*Sin[b*x], x] ==
(-I/2)*(x^(1 + m)*(-((a - I*b)*x))^(-1 - m)* Gamma[1 + m, -((a - I*b)*x)] - x^(1 + m)*(-((a + I*b)*x))^(-1 - m)* Gamma[1 + m, -((a + I*b)*x)]) 

           a x  m
Integrate[E    x  Sin[b x], x] == 

-I   1 + m                 -1 - m
-- (x      (-((a - I b) x))       
2
 
     Gamma[1 + m, -((a - I b) x)] - 
 
     1 + m                 -1 - m
    x      (-((a + I b) x))       
 
     Gamma[1 + m, -((a + I b) x)])
Time to compute: 0.16 second
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