3.29.45 \(\int \frac {1}{\sqrt {1-x} \sqrt {2+x} \sqrt {3+x}} \, dx\) [2845]

Optimal. Leaf size=18 \[ 2 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {2+x}}{\sqrt {3}}\right )\right |-3\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {120} \begin {gather*} 2 F\left (\left .\text {ArcSin}\left (\frac {\sqrt {x+2}}{\sqrt {3}}\right )\right |-3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*EllipticF[ArcSin[Sqrt[2 + x]/Sqrt[3]], -3]

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 &=2 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {2+x}}{\sqrt {3}}\right )\right |-3\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.25, size = 78, normalized size = 4.33 \begin {gather*} -\frac {2 i \sqrt {-((-1+x) (2+x))} \sqrt {3+x} F\left (i \sinh ^{-1}\left (\frac {\sqrt {3}}{\sqrt {-1+x}}\right )|\frac {4}{3}\right )}{\sqrt {3+\frac {9}{-1+x}} (-1+x)^{3/2} \sqrt {\frac {3+x}{-1+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 + x)*(2 + x))]*Sqrt[3 + x]*EllipticF[I*ArcSinh[Sqrt[3]/Sqrt[-1 + x]], 4/3])/(Sqrt[3 + 9/(-1
+ x)]*(-1 + x)^(3/2)*Sqrt[(3 + x)/(-1 + x)])

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Maple [A]
time = 0.10, size = 21, normalized size = 1.17

method	result	size
default	\(\frac {\EllipticF \left (\sqrt {3+x}, \frac {1}{2}\right ) \sqrt {-2-x}}{\sqrt {2+x}}\)	\(21\)
elliptic	\(\frac {\sqrt {-\left (-1+x \right ) \left (2+x \right ) \left (3+x \right )}\, \sqrt {-2-x}\, \EllipticF \left (\sqrt {3+x}, \frac {1}{2}\right )}{\sqrt {2+x}\, \sqrt {-x^{3}-4 x^{2}-x +6}}\)	\(51\)

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((3+x)^(1/2),1/2)*(-2-x)^(1/2)/(2+x)^(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 [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="fricas")

[Out]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {1 - x} \sqrt {x + 2} \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(x + 2)*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 = -0.06 \begin {gather*} \int \frac {1}{\sqrt {1-x}\,\sqrt {x+2}\,\sqrt {x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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