Solving $\int\left(\sqrt\frac{gx}{a\ln(x)+d+cx}\right)dx$ using Risch algorithm

Asked Aug 08 '19 at 05:57

Active Aug 09 '19 at 08:16

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I was solving some problems in particle physics, and i got stuck with such an equation,

$$\int\left(\sqrt\frac{gx}{a\ln(x)+b+cx}\right)dx$$

The problem was to formulate a function of time and displacement of a particle colliding with it's antiparticle with considering the Uehling correction of the Coulomb potential. After many steps i got stuck with the equation above. I've no idea how to solve this equation, i tried many times and i always failed. $a,b,c$ and $g$ are constants that i substituted with other complicated formulas in my equation so that the equation don't look huge.

I've heard of a method called the Risch algorithm, but i don't understand it really much, maybe this will be the key somehow.

Can some one please help me out with this equation. Thank you all guys.

edited Aug 09 '19 at 08:16

Community

asked Aug 08 '19 at 05:57

A.M.M Elsayed 马克

Integrals which are asked in books or in exams are do-able others may or not be so; clarify this point in the question. Also you could have written it as $\int \sqrt{\frac{x}{p \ln x +qx+r}} dx$ to receive more attention – Z Ahmed Aug 08 '19 at 06:09
no it's not in books or exams, i was thinking of a problem in physics and this equation suddenly came up, and i don't know what to do. Whatever, thank you for you suggestion @DrZafarAhmedDSc, i will edit the equation – A.M.M Elsayed 马克 Aug 08 '19 at 06:15
1

Better, mention the whole context here itself. People will appreciate it more. – Z Ahmed Aug 08 '19 at 06:18
ok@DrZafarAhmedDSc – A.M.M Elsayed 马克 Aug 08 '19 at 06:19
I think this is the work for the most general applications of the Risch algorithm, and as such, integrability may depend on the values of the parameters $a,b,c,g$. – orion Aug 08 '19 at 06:46
Are there any other ways?@orion – A.M.M Elsayed 马克 Aug 08 '19 at 06:54
If, as suggested, you consider $\int \sqrt{\frac{x}{p \ln x +qx+r}} dx$, I doubt about closed form solution if $q\ne 0$ – Claude Leibovici Aug 08 '19 at 06:59

Solving $\int\left(\sqrt\frac{gx}{a\ln(x)+d+cx}\right)dx$ using Risch algorithm

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