∫xln(a/x+rx) dx
=(1/2) ∫ln(a/x+rx) d(x^2)
=(1/2)x^2.ln(a/x+rx) -(1/2)∫[x^2/(a/x+rx)] ( -a/x^2 +r) dx
=(1/2)x^2.ln(a/x+rx) -(1/2)∫[ x(-a+rx^2)/(a+rx^2)] dx
=(1/2)x^2.ln(a/x+rx) -(1/2)∫ [x - 2ax/(a+rx^2)] dx
=(1/2)x^2.ln(a/x+rx) -(1/2)∫x dx + (1/2)∫ 2ax/(a+rx^2)] dx
=(1/2)x^2.ln(a/x+rx) -(1/4)x^2 + [1/(2r)]ln|a+rx^2| + C
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