3.9.70 \(\int \frac {\sqrt {-1+\frac {1}{x}} \sqrt {\frac {1}{x}} \sqrt {x}}{\sqrt {1+x}} \, dx\) [870]

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

[Out]

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

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).
time = 0.01, 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, 446, 112, 111} \begin {gather*} -\frac {2 \sqrt {\frac {1}{x}-1} \sqrt {\frac {1}{x}} \sqrt {-x} \sqrt {x} E\left (\left .\text {ArcSin}\left (\sqrt {-x}\right )\right |-1\right )}{\sqrt {1-x}} \end {gather*}

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 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], 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 112

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 446

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]

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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.03, size = 66, normalized size = 2.75 \begin {gather*} -\frac {2 \sqrt {\frac {x}{1+x}} \sqrt {1-x^2} \left (-3 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^2\right )+x \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};x^2\right )\right )}{3 \sqrt {1-x}} \end {gather*}

Antiderivative was successfully verified.

[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] Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(18)=36\).
time = 0.10, size = 49, normalized size = 2.04

method	result	size
default	\(-\frac {2 \sqrt {\frac {1}{x}}\, \sqrt {x}\, \sqrt {-\frac {-1+x}{x}}\, \EllipticE \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x}\, \sqrt {2-2 x}}{-1+x}\)	\(49\)
derivativedivides	\(-\frac {2 x^{\frac {5}{2}} \left (\frac {1}{x}\right )^{\frac {5}{2}} \sqrt {\left (\frac {1}{x}+1\right ) x}\, \sqrt {-1+\frac {1}{x}}\, \left (\sqrt {\frac {1}{x}+1}\, \sqrt {2}\, \sqrt {1-\frac {1}{x}}\, \sqrt {-\frac {1}{x}}\, \EllipticF \left (\sqrt {\frac {1}{x}+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\frac {1}{x}+1}\, \sqrt {2}\, \sqrt {1-\frac {1}{x}}\, \sqrt {-\frac {1}{x}}\, \EllipticE \left (\sqrt {\frac {1}{x}+1}, \frac {\sqrt {2}}{2}\right )-\frac {1}{x^{2}}+1\right )}{\frac {1}{x^{2}}-1}\)	\(122\)

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,method=_RETURNVERBOSE)

[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)*(2-2*x)^(1/2)/(-1+x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x} \sqrt {-1 + \frac {1}{x}} \sqrt {\frac {1}{x}}}{\sqrt {x + 1}}\, dx \end {gather*}

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]

Integral(sqrt(x)*sqrt(-1 + 1/x)*sqrt(1/x)/sqrt(x + 1), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {x}\,\sqrt {\frac {1}{x}-1}\,\sqrt {\frac {1}{x}}}{\sqrt {x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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