3.433 \(\int \frac {\sqrt {1-x}}{\sqrt {-x-x^2}} \, dx\)

Optimal. Leaf size=12 \[ -2 E\left (\left .\sin ^{-1}\left (\sqrt {-x}\right )\right |-1\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {714, 110} \[ -2 E\left (\left .\sin ^{-1}\left (\sqrt {-x}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x]/Sqrt[-x - x^2],x]

[Out]

-2*EllipticE[ArcSin[Sqrt[-x]], -1]

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]

Rule 714

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1
+ (c*x)/b]), 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 {-x} \sqrt {1+x}} \, dx\\ &=-2 E\left (\left .\sin ^{-1}\left (\sqrt {-x}\right )\right |-1\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 66, normalized size = 5.50 \[ -\frac {2 x \sqrt {1-x^2} \left (x \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};x^2\right )-3 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^2\right )\right )}{3 \sqrt {1-x} \sqrt {-x (x+1)}} \]

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*S
qrt[1 - x]*Sqrt[-(x*(1 + x))])

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fricas [F]  time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{2} - x} \sqrt {-x + 1}}{x^{2} + x}, x\right ) \]

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]

integral(-sqrt(-x^2 - x)*sqrt(-x + 1)/(x^2 + x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x + 1}}{\sqrt {-x^{2} - x}}\,{d x} \]

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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maple [B]  time = 0.06, size = 50, normalized size = 4.17 \[ \frac {2 \left (-x +1\right ) \sqrt {-\left (x +1\right ) x}\, \sqrt {x +1}\, \sqrt {2}\, \sqrt {-x}\, \EllipticE \left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )}{\left (x^{2}-1\right ) x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x + 1}}{\sqrt {-x^{2} - x}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {\sqrt {1-x}}{\sqrt {-x^2-x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1 - x}}{\sqrt {- x \left (x + 1\right )}}\, dx \]

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(1 - x)/sqrt(-x*(x + 1)), x)

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