3.9.66 \(\int \frac {\sqrt {1+x}}{\sqrt {x-x^2}} \, dx\) [866]

Optimal. Leaf size=10 \[ 2 E\left (\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {728, 111} \begin {gather*} 2 E\left (\left .\text {ArcSin}\left (\sqrt {x}\right )\right |-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x]/Sqrt[x - x^2],x]

[Out]

2*EllipticE[ArcSin[Sqrt[x]], -1]

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 728

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1
+ (c/b)*x]), x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4] && LtQ[
c, 0] && RationalQ[b]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.01, size = 64, normalized size = 6.40 \begin {gather*} \frac {2 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) x)} \sqrt {1+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x]/Sqrt[x - x^2],x]

[Out]

(2*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*Sqr
t[-((-1 + x)*x)]*Sqrt[1 + x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(8)=16\).
time = 0.08, size = 56, normalized size = 5.60

method	result	size
default	\(-\frac {2 \left (\EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )-\EllipticE \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {-x}\, \sqrt {2-2 x}\, \sqrt {-x \left (-1+x \right )}}{x \left (-1+x \right )}\)	\(56\)
elliptic	\(\frac {\sqrt {-x \left (x^{2}-1\right )}\, \left (\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-x^{3}+x}}+\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \left (-2 \EllipticE \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {-x^{3}+x}}\right )}{\sqrt {1+x}\, \sqrt {-x \left (-1+x \right )}}\)	\(116\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)^(1/2)/(-x^2+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

[Out]

integrate(sqrt(x + 1)/sqrt(-x^2 + x), 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+x)^(1/2)/(-x^2+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 + 1}}{\sqrt {- x \left (x - 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(sqrt(x + 1)/sqrt(-x^2 + x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.10 \begin {gather*} \int \frac {\sqrt {x+1}}{\sqrt {x-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(1/2)/(x - x^2)^(1/2),x)

[Out]

int((x + 1)^(1/2)/(x - x^2)^(1/2), x)

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