Optimal. Leaf size=57 \[ -\frac {2 \sqrt {x+1} \sqrt {x+2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {x}{5}+\frac {2}{5}}}\right ),\frac {1}{5}\right )}{\sqrt {5} \sqrt {-x-2} \sqrt {-x-1}} \]
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Rubi [A] time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {121, 118} \[ -\frac {2 \sqrt {x+1} \sqrt {x+2} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {x}{5}+\frac {2}{5}}}\right )|\frac {1}{5}\right )}{\sqrt {5} \sqrt {-x-2} \sqrt {-x-1}} \]
Antiderivative was successfully verified.
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Rule 118
Rule 121
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2-x} \sqrt {-1-x} \sqrt {-3+x}} \, dx &=\frac {\sqrt {2+x} \int \frac {1}{\sqrt {-1-x} \sqrt {\frac {2}{5}+\frac {x}{5}} \sqrt {-3+x}} \, dx}{\sqrt {5} \sqrt {-2-x}}\\ &=\frac {\left (\sqrt {1+x} \sqrt {2+x}\right ) \int \frac {1}{\sqrt {\frac {2}{5}+\frac {x}{5}} \sqrt {\frac {1}{4}+\frac {x}{4}} \sqrt {-3+x}} \, dx}{2 \sqrt {5} \sqrt {-2-x} \sqrt {-1-x}}\\ &=-\frac {2 \sqrt {1+x} \sqrt {2+x} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {2}{5}+\frac {x}{5}}}\right )|\frac {1}{5}\right )}{\sqrt {5} \sqrt {-2-x} \sqrt {-1-x}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 69, normalized size = 1.21 \[ \frac {i \sqrt {\frac {4}{x-3}+1} \sqrt {\frac {5}{x-3}+1} (x-3) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {2}{\sqrt {x-3}}\right ),\frac {5}{4}\right )}{\sqrt {-x-2} \sqrt {-x-1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x - 3} \sqrt {-x - 1} \sqrt {-x - 2}}{x^{3} - 7 \, x - 6}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x - 3} \sqrt {-x - 1} \sqrt {-x - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 46, normalized size = 0.81 \[ \frac {\sqrt {x +2}\, \sqrt {-x +3}\, \sqrt {x -3}\, \sqrt {-x -2}\, \EllipticF \left (\sqrt {-x -1}, \frac {i}{2}\right )}{x^{2}-x -6} \]
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 \[ \int \frac {1}{\sqrt {x - 3} \sqrt {-x - 1} \sqrt {-x - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {-x-1}\,\sqrt {-x-2}\,\sqrt {x-3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- x - 2} \sqrt {- x - 1} \sqrt {x - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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