Optimal. Leaf size=37 \[ \frac{x \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},1+\frac{1}{n};2+\frac{1}{n};-x^n\right )}{n+1} \]
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Rubi [A] time = 0.0314402, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{x \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},1+\frac{1}{n};2+\frac{1}{n};-x^n\right )}{n+1} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a*x^(2*n)]/Sqrt[1 + x^n],x]
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Rubi in Sympy [A] time = 8.88256, size = 36, normalized size = 0.97 \[ \frac{x^{- n} x^{n + 1} \sqrt{a x^{2 n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{n + 1}{n} \\ 2 + \frac{1}{n} \end{matrix}\middle |{- x^{n}} \right )}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*x**(2*n))**(1/2)/(1+x**n)**(1/2),x)
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Mathematica [A] time = 0.0489049, size = 53, normalized size = 1.43 \[ \frac{2 a x^{n+1} \left (\sqrt{x^n+1}-\, _2F_1\left (\frac{1}{2},\frac{1}{n};1+\frac{1}{n};-x^n\right )\right )}{(n+2) \sqrt{a x^{2 n}}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a*x^(2*n)]/Sqrt[1 + x^n],x]
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Maple [F] time = 0.102, size = 0, normalized size = 0. \[ \int{1\sqrt{a{x}^{2\,n}}{\frac{1}{\sqrt{1+{x}^{n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*x^(2*n))^(1/2)/(1+x^n)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{2 \, n}}}{\sqrt{x^{n} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x^(2*n))/sqrt(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(a*x^(2*n))/sqrt(x^n + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{2 n}}}{\sqrt{x^{n} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x**(2*n))**(1/2)/(1+x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{2 \, n}}}{\sqrt{x^{n} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x^(2*n))/sqrt(x^n + 1),x, algorithm="giac")
[Out]