Start by completing the square:
∫ 15 ln(x^2 - x + 4) dx
= ∫ 15 ln((x - 1/2)^2 + 15/4) dx
= ∫ 15 ln(w^2 + 15/4) dw, letting w = x - 1/2.
Now, use integration by parts with
u = 15 ln(w^2 + 15/4), dv = 1 dw
du = 30w dw/(w^2 + 15/4), v = w:
15w ln(w^2 + 15/4) - ∫ 30w^2 dw/(w^2 + 15/4)
= 15w ln(w^2 + 15/4) - 30 ∫ ((w^2 + 15/4) - 15/4) dw/(w^2 + 15/4)
= 15w ln(w^2 + 15/4) - 30 ∫ [1 - (15/4)/(w^2 + 15/4)] dw
= 15w ln(w^2 + 15/4) - 30w + (225/2) ∫ dw/(w^2 + 15/4)
= 15w ln(w^2 + 15/4) - 30w + (225/2) * (1/(√(15/4)) arctan(w / √(15/4)) + C
= 15w ln(w^2 + 15/4) - 30w + 15√15 arctan(2w/√15) + C
= 15(x - 1/2) ln((x - 1/2)^2 + 15/4) - 30(x - 1/2) + 15√15 arctan(2(x - 1/2)/√15) + C
= (15/2)(2x - 1) ln(x^2 - x + 4) - 15(2x - 1) + 15√15 arctan((2x - 1)/√15) + C.
I hope this helps!