Optimal. Leaf size=92 \[ -\frac{\sqrt{x+2} \sqrt{x+3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2}{\sqrt{x+3}}\right ),\frac{1}{4}\right )}{\sqrt{-x-3} \sqrt{-x-2}}-\frac{2 i K(4) \sqrt{x+2}}{\sqrt{-x-2}}+\frac{K\left (\frac{3}{4}\right ) \sqrt{x+3}}{\sqrt{-x-3}} \]
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Rubi [A] time = 0.0180816, antiderivative size = 52, normalized size of antiderivative = 0.57, 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} \[ -\frac{\sqrt{x+2} \sqrt{x+3} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{4}+\frac{3}{4}}}\right )|\frac{1}{4}\right )}{\sqrt{-x-3} \sqrt{-x-2}} \]
Warning: Unable to verify antiderivative.
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Rule 121
Rule 118
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3-x} \sqrt{-2-x} \sqrt{-1+x}} \, dx &=\frac{\sqrt{3+x} \int \frac{1}{\sqrt{-2-x} \sqrt{\frac{3}{4}+\frac{x}{4}} \sqrt{-1+x}} \, dx}{2 \sqrt{-3-x}}\\ &=\frac{\left (\sqrt{2+x} \sqrt{3+x}\right ) \int \frac{1}{\sqrt{\frac{3}{4}+\frac{x}{4}} \sqrt{\frac{2}{3}+\frac{x}{3}} \sqrt{-1+x}} \, dx}{2 \sqrt{3} \sqrt{-3-x} \sqrt{-2-x}}\\ &=-\frac{\sqrt{2+x} \sqrt{3+x} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{3}{4}+\frac{x}{4}}}\right )|\frac{1}{4}\right )}{\sqrt{-3-x} \sqrt{-2-x}}\\ \end{align*}
Mathematica [A] time = 0.087631, size = 75, normalized size = 0.82 \[ \frac{2 i \sqrt{\frac{3}{x-1}+1} \sqrt{\frac{4}{x-1}+1} (x-1) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{3}}{\sqrt{x-1}}\right ),\frac{4}{3}\right )}{\sqrt{-3 (x-1)-12} \sqrt{-x-2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 54, normalized size = 0.6 \begin{align*}{\frac{2\,\sqrt{3}}{3\,{x}^{2}+6\,x-9}{\it EllipticF} \left ( \sqrt{-2-x},{\frac{i}{3}}\sqrt{3} \right ) \sqrt{3+x}\sqrt{1-x}\sqrt{-1+x}\sqrt{-3-x}} \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 - 2} \sqrt{-x - 3}}\,{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 - 2} \sqrt{-x - 3}}{x^{3} + 4 \, x^{2} + 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 - 3} \sqrt{- x - 2} \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 - 2} \sqrt{-x - 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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