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I need to solve the definite integral: $$\int_{0}^{2} \sqrt{(1+x)\sqrt{4x+1}-3x+1}dx$$ The integral was proposed by my algebraic geometry professor as a warm up excercise, he hinted us to research about elliptic functions and curves, but I cannot find anything related to this integral. I tried substition and integration by parts, by I can't seem to reduce the problem.

I already solved the integral already by proving the convergence of it Taylor Series around $x_0=2$ and integrating said power series. But, its only an approximation, since I can't integrate infinite terms.

Is it possible to solve it analytically? What am I missing?

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    $\begingroup$ Please edit your query to show all of your work, including failed efforts down paths that didn't work. Please do not respond with a comment. $\endgroup$ Commented Apr 5, 2021 at 21:33
  • $\begingroup$ I couldn't find an analytic form, but numerical integration yields $3.23568$, which you might use to check an eventual answer. $\endgroup$ Commented Apr 5, 2021 at 21:39
  • $\begingroup$ Might be helpful to work with arcsin: math.stackexchange.com/questions/533082/… $\endgroup$ Commented Apr 5, 2021 at 22:02
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    $\begingroup$ As far as I got, with $4x+1=(u+1)^2$ we get $$\frac{1}{4}\int_0^2(u+1)\sqrt{u^3+8}\:du$$ $\endgroup$ Commented Apr 5, 2021 at 22:54
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    $\begingroup$ @ClaudeLeibovici: I hope some expert in elliptic curves can provide us a with suitable rational substitution to get the integral evaluated in terms of gamma functions. But this is something a professor shouldn't ask students as a warm up exercise. At least I am not getting warmed up. $\endgroup$ Commented Apr 6, 2021 at 5:24

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For a nightmare, look at the antiderivative given by Wolfram Alpha. If you like elliptic integrals and complex numbers, I suppose that you are more than happy.

Using the integration bounds, the result is just a monster. Numerically, $$I=3.23567625418595545329101113302458555831569511979865687\cdots$$

Interesting (for the fun) is that an inverse symbolic calculator proposes, as an approximation, the reciprocal of the smallest root of the cubic equation $$15572 x^3+13982 x^2-78333 x+22414=0$$

This is in a relative error of $3.1\times 10^{-19}$.

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