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Two projective algebraic curves are birationally equivalent if and only if they are isomorphic (Ch 1, Sec 6 of Hartshorne). You can input both of the above curves into Sage and compute various isomorphism class invariants to see that the curves are not isomorphic over C, hence not birationally equivalent. This paper http://www.warwick.ac.uk/~masjap/amsigusa.pdf has some basic facts about Igusa invariants in it.
sage: R.<x> = QQ[]
sage: C = HyperellipticCurve(x^6 + 14*x^4 + 5*x^3 + 14*x^2 + 18)
sage: D = HyperellipticCurve(x^6 + 14*x^4 + 5*x^3 + 14*x^2 + 5*x + 1)
sage: C.igusa_clebsch_invariants()
(-116896, 396482560, -4236950421504, -19899663217714003968)
sage: D.igusa_clebsch_invariants()
(-51616, 79673344, -1252989108224, -21978176867532800)
