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

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

[Out]

2*EllipticE[ArcSin[Sqrt[x]], -1]

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Rubi [A]  time = 0.0091033, antiderivative size = 10, normalized size of antiderivative = 1., 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 = {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{1-x} \sqrt{x}} \, dx\\ &=2 E\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.0111151, size = 64, normalized size = 6.4 \[ \frac{2 x \sqrt{1-x^2} \left (3 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^2\right )+x \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};x^2\right )\right )}{3 \sqrt{-(x-1) x} \sqrt{x+1}} \]

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*Sqr
t[-((-1 + x)*x)]*Sqrt[1 + x])

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Maple [B]  time = 0.007, size = 56, normalized size = 5.6 \begin{align*} -2\,{\frac{ \left ({\it EllipticF} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) -{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) \right ) \sqrt{-x}\sqrt{-2\,x+2}\sqrt{-x \left ( -1+x \right ) }}{x \left ( -1+x \right ) }} \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*(EllipticF((1+x)^(1/2),1/2*2^(1/2))-EllipticE((1+x)^(1/2),1/2*2^(1/2)))*(-x)^(1/2)*(-2*x+2)^(1/2)*(-x*(-1+x
))^(1/2)/x/(-1+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{x + 1}}{\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(x + 1)/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)