Calculating Integral using Pade Approximant & Fraction Expansion

[zetafunction]

i want to calculate the integral

$$\int_{4}^{\infty}dx \frac{x^{1/2}}{(x+3)^{5}}$$

my problem here is the square root of 'x' , my idea to overcome this and compute an approximation to the integral is to expand

$$x^{1/2} \approx \frac{P(x)}{Q(x)}$$ where P and Q are Polynomials, this can be made by using a Pade approximant for the the square root of 'x' or ifrom the identity

$$(x^{1/2}+1)(x^{1/2}+1)=x-1$$ expanding it into a finite continued fraction

so the integral $$\int_{4}^{\infty}dx \frac{P(x)}{(x+3)^{5}Q(x)}$$ can be easier computed by partial fraction expansion

can it be done ? i mean if Pade approximations can be valid on the entire real line (or at least for the points where x >0 )

 

[Dickfore]

Your integral is of the differential binomial type http://myyn.org/m/article/integration-of-differential-binomial/" . You can rationalize it exactly.

 

[SammyS]

... or from the identity

$$(x^{1/2}+1)(x^{1/2}+1)=x-1$$ expanding it into a finite continued fraction

so the integral ...

Your identity should be:

$$(x^{1/2}-1)(x^{1/2}+1)=x-1\ .$$