Optimal. Leaf size=41 \[ -\frac{2 \sqrt{x+2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{3}+\frac{2}{3}}}\right ),\frac{5}{3}\right )}{\sqrt{3} \sqrt{-x-2}} \]
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Rubi [A] time = 0.0107378, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {121, 118} \[ -\frac{2 \sqrt{x+2} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{3}+\frac{2}{3}}}\right )|\frac{5}{3}\right )}{\sqrt{3} \sqrt{-x-2}} \]
Antiderivative was successfully verified.
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Rule 121
Rule 118
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-2-x} \sqrt{-3+x} \sqrt{-1+x}} \, dx &=\frac{\sqrt{2+x} \int \frac{1}{\sqrt{\frac{2}{3}+\frac{x}{3}} \sqrt{-3+x} \sqrt{-1+x}} \, dx}{\sqrt{3} \sqrt{-2-x}}\\ &=-\frac{2 \sqrt{2+x} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{2}{3}+\frac{x}{3}}}\right )|\frac{5}{3}\right )}{\sqrt{3} \sqrt{-2-x}}\\ \end{align*}
Mathematica [C] time = 0.102355, size = 72, normalized size = 1.76 \[ \frac{2 i \sqrt{\frac{x-3}{x-1}} \sqrt{\frac{x-1}{x+2}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{3}}{\sqrt{-x-2}}\right ),\frac{5}{3}\right )}{\sqrt{3} \sqrt{\frac{x-3}{x+2}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.036, size = 72, normalized size = 1.8 \begin{align*} -{\frac{2\,\sqrt{3}}{3\,{x}^{3}-6\,{x}^{2}-15\,x+18}\sqrt{-2-x}\sqrt{-3+x}\sqrt{-1+x}\sqrt{2+x}\sqrt{1-x}\sqrt{3-x}{\it EllipticF} \left ({\frac{1}{5}\sqrt{10+5\,x}},{\frac{\sqrt{15}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x - 1} \sqrt{x - 3} \sqrt{-x - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x - 1} \sqrt{x - 3} \sqrt{-x - 2}}{x^{3} - 2 \, x^{2} - 5 \, x + 6}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- x - 2} \sqrt{x - 3} \sqrt{x - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x - 1} \sqrt{x - 3} \sqrt{-x - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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