3.29.42 \(\int \frac {1}{\sqrt {3-x} \sqrt {1+x} \sqrt {2+x}} \, dx\) [2842]

Optimal. Leaf size=16 \[ 2 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {1+x}}{2}\right )\right |-4\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, 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+1}}{2}\right )\right |-4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*EllipticF[ArcSin[Sqrt[1 + x]/2], -4]

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

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Mathematica [C] Result contains complex when optimal does not.
time = 1.23, size = 74, normalized size = 4.62 \begin {gather*} \frac {i \sqrt {1+\frac {4}{-3+x}} \sqrt {1+\frac {5}{-3+x}} (-3+x)^{3/2} F\left (i \sinh ^{-1}\left (\frac {2}{\sqrt {-3+x}}\right )|\frac {5}{4}\right )}{\sqrt {-((-3+x) (1+x))} \sqrt {2+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I*Sqrt[1 + 4/(-3 + x)]*Sqrt[1 + 5/(-3 + x)]*(-3 + x)^(3/2)*EllipticF[I*ArcSinh[2/Sqrt[-3 + x]], 5/4])/(Sqrt[-
((-3 + x)*(1 + x))]*Sqrt[2 + x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs. \(2(12)=24\).
time = 0.10, size = 29, normalized size = 1.81

method	result	size
default	\(\frac {2 \sqrt {5}\, \sqrt {-1-x}\, \EllipticF \left (\sqrt {2+x}, \frac {\sqrt {5}}{5}\right )}{5 \sqrt {1+x}}\)	\(29\)
elliptic	\(\frac {2 \sqrt {-\left (-3+x \right ) \left (1+x \right ) \left (2+x \right )}\, \sqrt {15-5 x}\, \sqrt {-1-x}\, \EllipticF \left (\sqrt {2+x}, \frac {\sqrt {5}}{5}\right )}{5 \sqrt {3-x}\, \sqrt {1+x}\, \sqrt {-x^{3}+7 x +6}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5/(1+x)^(1/2)*5^(1/2)*(-1-x)^(1/2)*EllipticF((2+x)^(1/2),1/5*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/(3-x)^(1/2)/(1+x)^(1/2)/(2+x)^(1/2),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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