3.9.62 \(\int \frac {1}{\sqrt {1+x} \sqrt {x-x^2}} \, dx\) [862]

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

[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 = {728, 116} \begin {gather*} 2 F\left (\left .\text {ArcSin}\left (\sqrt {x}\right )\right |-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 + x]*Sqrt[x - x^2]),x]

[Out]

2*EllipticF[ArcSin[Sqrt[x]], -1]

Rule 116

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

Rule 728

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Int[(d + e*x)^m/(Sqrt[b*x]*Sqrt[1
+ (c/b)*x]), 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] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.01, size = 44, normalized size = 4.40 \begin {gather*} \frac {2 x \sqrt {1-x^2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^2\right )}{\sqrt {-((-1+x) x)} \sqrt {1+x}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs. \(2(8)=16\).
time = 0.10, size = 42, normalized size = 4.20

method	result	size
default	\(-\frac {\EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right ) \sqrt {-x}\, \sqrt {2-2 x}\, \sqrt {-x \left (-1+x \right )}}{x \left (-1+x \right )}\)	\(42\)
elliptic	\(\frac {\sqrt {-x \left (x^{2}-1\right )}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-x \left (-1+x \right )}\, \sqrt {-x^{3}+x}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+x)^(1/2)/(-x^2+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

0

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

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.10 \begin {gather*} \int \frac {1}{\sqrt {x-x^2}\,\sqrt {x+1}} \,d x \end {gather*}

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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