We have
sage: q = -1/17*sqrt(-52/17*I + 47/17)/(2*x - sqrt(4*I + 1) - 1)
....: print( integrate( q, x, 0, 1, algorithm='maxima' ).n() )
....: print( integrate( q, x, 0, 1, algorithm='sympy' ).n() )
....: print( integrate( q, x, 0, 1, algorithm='mathematica_free' ).n() )
0.0400005859357106 - 0.0177639491598775*I
0.0236417600209372 - 0.0546005003074790*I
0.0236418000000000 - 0.0546005000000000*I
Is this expected?