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

Optimal. Leaf size=48 \[ \frac{2 x \sqrt{a x^n} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (1+\frac{2}{n}\right );\frac{1}{2} \left (3+\frac{2}{n}\right );-x^n\right )}{n+2} \]

[Out]

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

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Rubi [A]  time = 0.0331198, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 x \sqrt{a x^n} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (1+\frac{2}{n}\right );\frac{1}{2} \left (3+\frac{2}{n}\right );-x^n\right )}{n+2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^n]/Sqrt[1 + x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 9.17569, size = 42, normalized size = 0.88 \[ \frac{2 x^{- \frac{n}{2}} x^{\frac{n}{2} + 1} \sqrt{a x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{n + 2}{2 n} \\ \frac{3}{2} + \frac{1}{n} \end{matrix}\middle |{- x^{n}} \right )}}{n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.027451, size = 40, normalized size = 0.83 \[ \frac{2 x \sqrt{a x^n} \, _2F_1\left (\frac{1}{2},\frac{1}{2}+\frac{1}{n};\frac{3}{2}+\frac{1}{n};-x^n\right )}{n+2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x^n]/Sqrt[1 + x^n],x]

[Out]

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

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Maple [A]  time = 0.063, size = 35, normalized size = 0.7 \[ 2\,{\frac{x{\mbox{$_2$F$_1$}(1/2,1/2+{n}^{-1};\,3/2+{n}^{-1};\,-{x}^{n})}\sqrt{a{x}^{n}}}{2+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^n)^(1/2)/(1+x^n)^(1/2),x)

[Out]

2*x*hypergeom([1/2,1/2+1/n],[3/2+1/n],-x^n)*(a*x^n)^(1/2)/(2+n)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a*x^n)/sqrt(x^n + 1), x)

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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^n)/sqrt(x^n + 1),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a*x**n)/sqrt(x**n + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a*x^n)/sqrt(x^n + 1), x)