Optimal. Leaf size=52 \[ \frac{4 x \sqrt{a x^{n/2}} \, _2F_1\left (\frac{1}{2},\frac{1}{4} \left (1+\frac{4}{n}\right );\frac{1}{4} \left (5+\frac{4}{n}\right );-x^n\right )}{n+4} \]
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Rubi [A] time = 0.0361213, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{4 x \sqrt{a x^{n/2}} \, _2F_1\left (\frac{1}{2},\frac{1}{4} \left (1+\frac{4}{n}\right );\frac{1}{4} \left (5+\frac{4}{n}\right );-x^n\right )}{n+4} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a*x^(n/2)]/Sqrt[1 + x^n],x]
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Rubi in Sympy [A] time = 9.07168, size = 44, normalized size = 0.85 \[ \frac{4 x^{- \frac{n}{4}} x^{\frac{n}{4} + 1} \sqrt{a x^{\frac{n}{2}}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{n + 4}{4 n} \\ \frac{5}{4} + \frac{1}{n} \end{matrix}\middle |{- x^{n}} \right )}}{n + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*x**(1/2*n))**(1/2)/(1+x**n)**(1/2),x)
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Mathematica [A] time = 0.0293149, size = 44, normalized size = 0.85 \[ \frac{4 x \sqrt{a x^{n/2}} \, _2F_1\left (\frac{1}{2},\frac{1}{4}+\frac{1}{n};\frac{5}{4}+\frac{1}{n};-x^n\right )}{n+4} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a*x^(n/2)]/Sqrt[1 + x^n],x]
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Maple [A] time = 0.089, size = 37, normalized size = 0.7 \[ 4\,{\frac{x{\mbox{$_2$F$_1$}(1/2,1/4+{n}^{-1};\,5/4+{n}^{-1};\,-{x}^{n})}\sqrt{a{x}^{n/2}}}{4+n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*x^(1/2*n))^(1/2)/(1+x^n)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{\frac{1}{2} \, n}}}{\sqrt{x^{n} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x^(1/2*n))/sqrt(x^n + 1),x, algorithm="maxima")
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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^(1/2*n))/sqrt(x^n + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{\frac{n}{2}}}}{\sqrt{x^{n} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x**(1/2*n))**(1/2)/(1+x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{\frac{1}{2} \, n}}}{\sqrt{x^{n} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a*x^(1/2*n))/sqrt(x^n + 1),x, algorithm="giac")
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