G'day wizards of Maple

Hello humble forum readers

I haven't used Maple in over 10 years and am therefor a bit rusty when it comes to runtime optimizations; on top of that, I'm not a mathematician, so bear with my limited knowledge in this field. A risk analysis problem of a rather complex system resulted in roughly following integral, which I seem to be unable to compute with Maple (Version 12, in case this matters):

binom[1]:=(a1+b1*exp(-(a1+b1)*t))^k[1];

binom[2]:=(a2+b2*exp(-(a2+b2)*t))^k[2];