3.1008 \(\int \sqrt{\frac{x^n}{1+x^n}} \, dx\)

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]

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

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Rubi [A]  time = 0.0159383, 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, Rules used = {1958, 15, 364} \[ \frac{2 x \sqrt{x^n} \, _2F_1\left (\frac{1}{2},\frac{1}{2} \left (1+\frac{2}{n}\right );\frac{1}{2} \left (3+\frac{2}{n}\right );-x^n\right )}{n+2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x^n/(1 + x^n)],x]

[Out]

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

Rule 1958

Int[(u_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[(u*(e*(a + b*x
^n))^p)/(c + d*x^n)^p, x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[b*d*e, 0] && GtQ[c - (a*d)/b, 0]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \sqrt{\frac{x^n}{1+x^n}} \, dx &=\int \frac{\sqrt{x^n}}{\sqrt{1+x^n}} \, dx\\ &=\left (x^{-n/2} \sqrt{x^n}\right ) \int \frac{x^{n/2}}{\sqrt{1+x^n}} \, dx\\ &=\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 )}{2+n}\\ \end{align*}

Mathematica [A]  time = 0.0119177, 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]

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

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{\frac{{x}^{n}}{1+{x}^{n}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^n/(1+x^n))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^n/(x^n + 1)), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^n/(1+x^n))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**n/(1+x**n))**(1/2),x)

[Out]

Integral(sqrt(x**n/(x**n + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{x^{n}}{x^{n} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^n/(1+x^n))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(x^n/(x^n + 1)), x)