3.29.41 \(\int \frac {1}{\sqrt {1+x} \sqrt {2+x} \sqrt {3+x}} \, dx\) [2841]

Optimal. Leaf size=12 \[ -2 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {3+x}}\right )\right |2\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {119} \begin {gather*} -2 F\left (\left .\text {ArcSin}\left (\frac {1}{\sqrt {x+3}}\right )\right |2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*EllipticF[ArcSin[1/Sqrt[3 + x]], 2]

Rule 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[-2*(Sqrt[d/
f]/(d*Rt[-(b*e - a*f)/f, 2]))*EllipticF[ArcSin[Rt[-(b*e - a*f)/f, 2]/Sqrt[a + b*x]], f*((b*c - a*d)/(d*(b*e -
a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[d/b, 0] && GtQ[f/b, 0] && LeQ[c, a*(d/b)] && LeQ[e, a*(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 {1}{\sqrt {3+x}}\right )\right |2\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.18, size = 55, normalized size = 4.58 \begin {gather*} \frac {2 i \sqrt {1+\frac {1}{1+x}} F\left (\left .i \sinh ^{-1}\left (\frac {1}{\sqrt {1+x}}\right )\right |2\right )}{\sqrt {\frac {2+x}{3+x}} \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 + (1 + x)^(-1)]*EllipticF[I*ArcSinh[1/Sqrt[1 + x]], 2])/(Sqrt[(2 + x)/(3 + x)]*Sqrt[(3 + x)/(1 +
x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(11)=22\).
time = 0.09, size = 50, normalized size = 4.17

method	result	size
default	\(\frac {\EllipticF \left (\sqrt {3+x}, \frac {\sqrt {2}}{2}\right ) \sqrt {-2-x}\, \sqrt {2}\, \sqrt {-1-x}\, \sqrt {2+x}\, \sqrt {1+x}}{x^{2}+3 x +2}\)	\(50\)
elliptic	\(\frac {\sqrt {\left (1+x \right ) \left (2+x \right ) \left (3+x \right )}\, \sqrt {-2-2 x}\, \sqrt {-2-x}\, \EllipticF \left (\sqrt {3+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {1+x}\, \sqrt {2+x}\, \sqrt {x^{3}+6 x^{2}+11 x +6}}\)	\(64\)

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^(1/2))*(-2-x)^(1/2)*2^(1/2)*(-1-x)^(1/2)*(2+x)^(1/2)*(1+x)^(1/2)/(x^2+3*x+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 = 0.17, size = 8, normalized size = 0.67 \begin {gather*} 2 \, {\rm weierstrassPInverse}\left (4, 0, x + 2\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(4, 0, x + 2)

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Sympy [C] Result contains complex when optimal does not.
time = 13.19, size = 65, normalized size = 5.42 \begin {gather*} - \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {1}{\left (x + 2\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {{G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {e^{2 i \pi }}{\left (x + 2\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \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]

-meijerg(((1/2, 1, 1), (3/4, 3/4, 5/4)), ((1/4, 1/2, 3/4, 1, 5/4), (0,)), (x + 2)**(-2))/(4*pi**(3/2)) + meije
rg(((-1/4, 0, 1/4, 1/2, 3/4), (1,)), ((0, 1/2, 0), (-1/4, 1/4, 1/4)), exp_polar(2*I*pi)/(x + 2)**2)/(4*pi**(3/
2))

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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 = -0.08 \begin {gather*} \int \frac {1}{\sqrt {x+1}\,\sqrt {x+2}\,\sqrt {x+3}} \,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)*(x + 3)^(1/2)),x)

[Out]

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

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