Optimal. Leaf size=10 \[ 2 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{x}\right ),-1\right ) \]
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Rubi [A] time = 0.0091625, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {714, 115} \[ 2 F\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right ) \]
Antiderivative was successfully verified.
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Rule 714
Rule 115
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+x} \sqrt{x-x^2}} \, dx &=\int \frac{1}{\sqrt{1-x} \sqrt{x} \sqrt{1+x}} \, dx\\ &=2 F\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right )\\ \end{align*}
Mathematica [C] time = 0.0070105, size = 44, normalized size = 4.4 \[ \frac{2 x \sqrt{1-x^2} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};x^2\right )}{\sqrt{-(x-1) x} \sqrt{x+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 42, normalized size = 4.2 \begin{align*} -{\frac{1}{x \left ( -1+x \right ) }{\it EllipticF} \left ( \sqrt{1+x},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-x}\sqrt{-2\,x+2}\sqrt{-x \left ( -1+x \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^{2} + x} \sqrt{x + 1}}\,{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^{2} + x} \sqrt{x + 1}}{x^{3} - x}, 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 \left (x - 1\right )} \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^{2} + x} \sqrt{x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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