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

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

[Out]

2*EllipticF(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 = {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 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])

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 {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*}

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Mathematica [C]  time = 0.01, size = 44, normalized size = 4.40 \[ \frac {2 x \sqrt {1-x^2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^2\right )}{\sqrt {-((x-1) x)} \sqrt {x+1}} \]

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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fricas [F]  time = 1.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{2} + x} \sqrt {x + 1}}{x^{3} - x}, x\right ) \]

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

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)

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maple [B]  time = 0.01, size = 43, normalized size = 4.30 \[ \frac {\sqrt {-x}\, \sqrt {-2 x +2}\, \sqrt {-\left (x -1\right ) x}\, \EllipticF \left (\sqrt {x +1}, \frac {\sqrt {2}}{2}\right )}{\left (-x +1\right ) x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.10 \[ \int \frac {1}{\sqrt {x-x^2}\,\sqrt {x+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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