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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]