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Integrate[Log[a*x]*Log[a + b*x], x] ==
(2*b*x - a*Log[a + b*x] - b*x*Log[a + b*x] + Log[a*x]*(-(b*x) + b*x*Log[a + b*x] + a*Log[1 + (b*x)/a]) + a*PolyLog[2, -((b*x)/a)])/b 

Integrate[Log[a x] Log[a + b x], x] == 

(2 b x - a Log[a + b x] - b x Log[a + b x] + 
 
    Log[a x] (-(b x) + b x Log[a + b x] + 
 
                 b x                    b x
       a Log[1 + ---]) + a PolyLog[2, -(---)]) / b
                  a                      a
Time to compute: 0.06 second
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