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Question Number 108839 by 150505R last updated on 19/Aug/20

Commented by Her_Majesty last updated on 20/Aug/20

this looks to me as if the integral must be ((p(x^2 +x+1)^(3/2) )/x^q ) with p, q ∈Q (d/dx)[(((x^2 +x+1)^(3/2) p)/x^q )]=−((p(2(q−3)x^2 +(2q−3)x+2q)(√(x^2 +x+1)))/x^(q+1) ) ⇒ q=9/2 −((3p(x^2 +2x+3)(√(x^2 +x+1)))/(2x^(11/2) )) ⇒ p=−2/3 ⇒ ∫(((x^2 +2x+3)(√(x^2 +x+1)))/x^(11/2) )dx=−((2(x^2 +x+1)^(3/2) )/(3x^(9/2) ))+C

thislookstomeasiftheintegralmustbep(x2+x+1)3/2xqwithp,q∈Qddx[(x2+x+1)32pxq]=−p(2(q−3)x2+(2q−3)x+2q)x2+x+1xq+1⇒q=9/2−3p(x2+2x+3)x2+x+12x11/2⇒p=−2/3⇒∫(x2+2x+3)x2+x+1x11/2dx=−2(x2+x+1)3/23x9/2+C

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