Optimal. Leaf size=12 \[ -2 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {3+x}}\right )\right |2\right ) \]
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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.
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Rule 119
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.
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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.
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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.
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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.
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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.
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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.
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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.
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