3.870 \(\int \sqrt{\frac{x^n}{1+x^n}} \, dx\)

Optimal. Leaf size=46 \[ \frac{2 x \sqrt{x^n} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (1+\frac{2}{n}\right );\frac{1}{2} \left (3+\frac{2}{n}\right );-x^n\right )}{n+2} \]

[Out]

(2*x*Sqrt[x^n]*Hypergeometric2F1[1/2, (1 + 2/n)/2, (3 + 2/n)/2, -x^n])/(2 + n)

_______________________________________________________________________________________

Rubi [A]  time = 0.0376636, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 x \sqrt{x^n} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (1+\frac{2}{n}\right );\frac{1}{2} \left (3+\frac{2}{n}\right );-x^n\right )}{n+2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x^n/(1 + x^n)],x]

[Out]

(2*x*Sqrt[x^n]*Hypergeometric2F1[1/2, (1 + 2/n)/2, (3 + 2/n)/2, -x^n])/(2 + n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.21582, size = 41, normalized size = 0.89 \[ \frac{2 x^{- \frac{n}{2}} x^{\frac{n}{2} + 1} \sqrt{x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{n + 2}{2 n} \\ \frac{3}{2} + \frac{1}{n} \end{matrix}\middle |{- x^{n}} \right )}}{n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**n/(1+x**n))**(1/2),x)

[Out]

2*x**(-n/2)*x**(n/2 + 1)*sqrt(x**n)*hyper((1/2, (n + 2)/(2*n)), (3/2 + 1/n,), -x
**n)/(n + 2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0320127, size = 38, normalized size = 0.83 \[ \frac{2 x \sqrt{x^n} \, _2F_1\left (\frac{1}{2},\frac{1}{2}+\frac{1}{n};\frac{3}{2}+\frac{1}{n};-x^n\right )}{n+2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x^n/(1 + x^n)],x]

[Out]

(2*x*Sqrt[x^n]*Hypergeometric2F1[1/2, 1/2 + n^(-1), 3/2 + n^(-1), -x^n])/(2 + n)

_______________________________________________________________________________________

Maple [F]  time = 0.027, size = 0, normalized size = 0. \[ \int \sqrt{{\frac{{x}^{n}}{1+{x}^{n}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^n/(1+x^n))^(1/2),x)

[Out]

int((x^n/(1+x^n))^(1/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x^{n}}{x^{n} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^n/(x^n + 1)),x, algorithm="maxima")

[Out]

integrate(sqrt(x^n/(x^n + 1)), x)

_______________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^n/(x^n + 1)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x^{n}}{x^{n} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**n/(1+x**n))**(1/2),x)

[Out]

Integral(sqrt(x**n/(x**n + 1)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x^{n}}{x^{n} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^n/(x^n + 1)),x, algorithm="giac")

[Out]

integrate(sqrt(x^n/(x^n + 1)), x)