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

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

[Out]

2*EllipticF[ArcSin[Sqrt[x]], -1]

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Rubi [A]  time = 0.0091625, 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, 115} \[ 2 F\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 + x]*Sqrt[x - x^2]),x]

[Out]

2*EllipticF[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 115

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d), 2]*E
llipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-(b/d), 2])], (c*f)/(d*e)])/(b*Sqrt[e]), x] /; FreeQ[{b, c, d, e, f}, x]
&& GtQ[c, 0] && GtQ[e, 0] && (GtQ[-(b/d), 0] || LtQ[-(b/f), 0])

Rubi steps

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

Mathematica [C]  time = 0.0070105, size = 44, normalized size = 4.4 \[ \frac{2 x \sqrt{1-x^2} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^2\right )}{\sqrt{-(x-1) x} \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[1 - x^2]*Hypergeometric2F1[1/4, 1/2, 5/4, x^2])/(Sqrt[-((-1 + x)*x)]*Sqrt[1 + x])

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Maple [B]  time = 0.006, size = 42, normalized size = 4.2 \begin{align*} -{\frac{1}{x \left ( -1+x \right ) }{\it EllipticF} \left ( \sqrt{1+x},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-x}\sqrt{-2\,x+2}\sqrt{-x \left ( -1+x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-EllipticF((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{1}{\sqrt{-x^{2} + x} \sqrt{x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)^(1/2)/(-x^2+x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^2 + x)*sqrt(x + 1)), 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^{3} - x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)^(1/2)/(-x^2+x)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-x^2 + x)*sqrt(x + 1)/(x^3 - x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+x)^(1/2)/(-x^2+x)^(1/2),x, algorithm="giac")

[Out]

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