3.232 \(\int \frac{\sqrt{a x^{2 n}}}{\sqrt{1+x^n}} \, dx\)

Optimal. Leaf size=37 \[ \frac{x \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},1+\frac{1}{n};2+\frac{1}{n};-x^n\right )}{n+1} \]

[Out]

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

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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]

[Out]

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

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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)

[Out]

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

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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]

[Out]

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

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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]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{2 \, n}}}{\sqrt{x^{n} + 1}}\,{d x} \]

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]

integrate(sqrt(a*x^(2*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^(2*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. \[ \int \frac{\sqrt{a x^{2 n}}}{\sqrt{x^{n} + 1}}\, dx \]

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]

Integral(sqrt(a*x**(2*n))/sqrt(x**n + 1), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{2 \, n}}}{\sqrt{x^{n} + 1}}\,{d x} \]

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]

integrate(sqrt(a*x^(2*n))/sqrt(x^n + 1), x)