Optimal. Leaf size=16 \[ 2 \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {x+1}}{2}\right ),-4\right ) \]
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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 = {119} \[ 2 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {x+1}}{2}\right )\right |-4\right ) \]
Antiderivative was successfully verified.
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Rule 119
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] time = 0.11, size = 74, normalized size = 4.62 \[ \frac {i \sqrt {\frac {4}{x-3}+1} \sqrt {\frac {5}{x-3}+1} (x-3)^{3/2} \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {2}{\sqrt {x-3}}\right ),\frac {5}{4}\right )}{\sqrt {-((x-3) (x+1))} \sqrt {x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x + 2} \sqrt {x + 1} \sqrt {-x + 3}}{x^{3} - 7 \, x - 6}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x + 2} \sqrt {x + 1} \sqrt {-x + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 1.56 \[ -\frac {\sqrt {-x -1}\, \EllipticF \left (\sqrt {-x -1}, \frac {i}{2}\right )}{\sqrt {x +1}} \]
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 \[ \int \frac {1}{\sqrt {x + 2} \sqrt {x + 1} \sqrt {-x + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {x+1}\,\sqrt {x+2}\,\sqrt {3-x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 - x} \sqrt {x + 1} \sqrt {x + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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