### 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[ArcSin[Sqrt[-x]], -1]

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Rubi [A]  time = 0.0104788, antiderivative size = 12, normalized size of antiderivative = 1., 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 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]

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-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*}

Mathematica [C]  time = 0.0115723, size = 66, normalized size = 5.5 $-\frac{2 x \sqrt{1-x^2} \left (x \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};x^2\right )-3 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^2\right )\right )}{3 \sqrt{1-x} \sqrt{-x (x+1)}}$

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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Maple [B]  time = 0.105, size = 38, normalized size = 3.2 \begin{align*} -2\,{\frac{{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) \sqrt{-x}\sqrt{2}\sqrt{-x \left ( 1+x \right ) }}{\sqrt{1+x}x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x + 1}}{\sqrt{-x^{2} - x}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{2} - x} \sqrt{-x + 1}}{x^{2} + x}, x\right ) \end{align*}

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

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x + 1}}{\sqrt{-x^{2} - x}}\,{d x} \end{align*}

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)