3.29.47 \(\int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3+x}} \, dx\) [2847]

Optimal. Leaf size=23 \[ \frac {2 F\left (\sin ^{-1}\left (\frac {\sqrt {3+x}}{2}\right )|\frac {4}{5}\right )}{\sqrt {5}} \]

[Out]

2/5*EllipticF(1/2*(3+x)^(1/2),2/5*5^(1/2))*5^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {120} \begin {gather*} \frac {2 F\left (\text {ArcSin}\left (\frac {\sqrt {x+3}}{2}\right )|\frac {4}{5}\right )}{\sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - x]*Sqrt[2 - x]*Sqrt[3 + x]),x]

[Out]

(2*EllipticF[ArcSin[Sqrt[3 + x]/2], 4/5])/Sqrt[5]

Rule 120

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3+x}} \, dx &=\frac {2 F\left (\sin ^{-1}\left (\frac {\sqrt {3+x}}{2}\right )|\frac {4}{5}\right )}{\sqrt {5}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.28, size = 65, normalized size = 2.83 \begin {gather*} -\frac {2 i \sqrt {1-\frac {4}{1-x}} \sqrt {1+\frac {1}{1-x}} (1-x) F\left (\left .i \sinh ^{-1}\left (\frac {1}{\sqrt {1-x}}\right )\right |-4\right )}{\sqrt {-((-2+x) (3+x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 - x]*Sqrt[2 - x]*Sqrt[3 + x]),x]

[Out]

((-2*I)*Sqrt[1 - 4/(1 - x)]*Sqrt[1 + (1 - x)^(-1)]*(1 - x)*EllipticF[I*ArcSinh[1/Sqrt[1 - x]], -4])/Sqrt[-((-2
+ x)*(3 + x))]

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Maple [A]
time = 0.15, size = 16, normalized size = 0.70

method	result	size
default	\(\EllipticF \left (\frac {\sqrt {15+5 x}}{5}, \frac {\sqrt {5}}{2}\right )\)	\(16\)
elliptic	\(\frac {\sqrt {\left (-1+x \right ) \left (-2+x \right ) \left (3+x \right )}\, \sqrt {15+5 x}\, \sqrt {10-5 x}\, \EllipticF \left (\frac {\sqrt {15+5 x}}{5}, \frac {\sqrt {5}}{2}\right )}{5 \sqrt {2-x}\, \sqrt {3+x}\, \sqrt {x^{3}-7 x +6}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(1/2)/(2-x)^(1/2)/(3+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

EllipticF(1/5*(15+5*x)^(1/2),1/2*5^(1/2))

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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)/(2-x)^(1/2)/(3+x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.19, size = 6, normalized size = 0.26 \begin {gather*} 2 \, {\rm weierstrassPInverse}\left (28, -24, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/2)/(2-x)^(1/2)/(3+x)^(1/2),x, algorithm="fricas")

[Out]

2*weierstrassPInverse(28, -24, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {1 - x} \sqrt {2 - x} \sqrt {x + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)**(1/2)/(2-x)**(1/2)/(3+x)**(1/2),x)

[Out]

Integral(1/(sqrt(1 - x)*sqrt(2 - x)*sqrt(x + 3)), 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)/(2-x)^(1/2)/(3+x)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{\sqrt {1-x}\,\sqrt {2-x}\,\sqrt {x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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