3.870 \(\int \frac{\sqrt{-1+\frac{1}{x}} \sqrt{\frac{1}{x}} \sqrt{x}}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=24 \[ -\frac{2 \sqrt{-x} E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )}{\sqrt{x}} \]

[Out]

(-2*Sqrt[-x]*EllipticE[ArcSin[Sqrt[-x]], -1])/Sqrt[x]

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Rubi [B]  time = 0.0153987, antiderivative size = 49, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {15, 435, 111, 110} \[ -\frac{2 \sqrt{\frac{1}{x}-1} \sqrt{\frac{1}{x}} \sqrt{-x} \sqrt{x} E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )}{\sqrt{1-x}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[-1 + x^(-1)]*Sqrt[x^(-1)]*Sqrt[x])/Sqrt[1 + x],x]

[Out]

(-2*Sqrt[-1 + x^(-1)]*Sqrt[x^(-1)]*Sqrt[-x]*Sqrt[x]*EllipticE[ArcSin[Sqrt[-x]], -1])/Sqrt[1 - x]

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 435

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(x^(n*FracPart[q])*(c +
d/x^n)^FracPart[q])/(d + c*x^n)^FracPart[q], Int[((a + b*x^n)^p*(d + c*x^n)^q)/x^(n*q), x], x] /; FreeQ[{a, b,
c, d, n, p, q}, x] && EqQ[mn, -n] &&  !IntegerQ[q] &&  !IntegerQ[p]

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[-(b*x)]/Sqrt[b*
x], Int[Sqrt[e + f*x]/(Sqrt[-(b*x)]*Sqrt[c + d*x]), x], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &
& GtQ[c, 0] && GtQ[e, 0] && LtQ[-(b/d), 0]

Rule 110

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*Sqrt[e]*Rt[-(b/d)
, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/b, x] /; FreeQ[{b, c, d, e, f}, x] &&
NeQ[d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-(b/d), 0]

Rubi steps

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

Mathematica [C]  time = 0.0397828, size = 66, normalized size = 2.75 \[ -\frac{2 \sqrt{\frac{x}{x+1}} \sqrt{1-x^2} \left (x \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};x^2\right )-3 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^2\right )\right )}{3 \sqrt{1-x}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[-1 + x^(-1)]*Sqrt[x^(-1)]*Sqrt[x])/Sqrt[1 + x],x]

[Out]

(-2*Sqrt[x/(1 + x)]*Sqrt[1 - x^2]*(-3*Hypergeometric2F1[1/4, 1/2, 5/4, x^2] + x*Hypergeometric2F1[1/2, 3/4, 7/
4, x^2]))/(3*Sqrt[1 - x])

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Maple [B]  time = 0.008, size = 49, normalized size = 2. \begin{align*} -2\,{\frac{\sqrt{{x}^{-1}}\sqrt{x}{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) \sqrt{-x}\sqrt{-2\,x+2}}{-1+x}\sqrt{-{\frac{-1+x}{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+1/x)^(1/2)*(1/x)^(1/2)*x^(1/2)/(1+x)^(1/2),x)

[Out]

-2*(1/x)^(1/2)*x^(1/2)*(-(-1+x)/x)^(1/2)*EllipticE((1+x)^(1/2),1/2*2^(1/2))*(-x)^(1/2)*(-2*x+2)^(1/2)/(-1+x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(1/x - 1)/sqrt(x + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-\frac{x - 1}{x}}}{\sqrt{x + 1}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-(x - 1)/x)/sqrt(x + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(1/x - 1)/sqrt(x + 1), x)