Optimal. Leaf size=27 \[ -2 \sqrt {-x^2+x+2}-2 \sin ^{-1}\left (\frac {1}{3} (1-2 x)\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {640, 619, 216} \[ -2 \sqrt {-x^2+x+2}-2 \sin ^{-1}\left (\frac {1}{3} (1-2 x)\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 619
Rule 640
Rubi steps
\begin {align*} \int \frac {1+2 x}{\sqrt {2+x-x^2}} \, dx &=-2 \sqrt {2+x-x^2}+2 \int \frac {1}{\sqrt {2+x-x^2}} \, dx\\ &=-2 \sqrt {2+x-x^2}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{9}}} \, dx,x,1-2 x\right )\\ &=-2 \sqrt {2+x-x^2}-2 \sin ^{-1}\left (\frac {1}{3} (1-2 x)\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ -2 \sqrt {-x^2+x+2}-2 \sin ^{-1}\left (\frac {1}{3} (1-2 x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 36, normalized size = 1.33 \[ -2 \sqrt {-x^2+x+2}-4 \tan ^{-1}\left (\frac {\sqrt {-x^2+x+2}}{x+1}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 43, normalized size = 1.59 \[ -2 \, \sqrt {-x^{2} + x + 2} - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + x + 2} {\left (2 \, x - 1\right )}}{2 \, {\left (x^{2} - x - 2\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 21, normalized size = 0.78 \[ -2 \, \sqrt {-x^{2} + x + 2} + 2 \, \arcsin \left (\frac {2}{3} \, x - \frac {1}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 22, normalized size = 0.81
method | result | size |
default | \(2 \arcsin \left (-\frac {1}{3}+\frac {2 x}{3}\right )-2 \sqrt {-x^{2}+x +2}\) | \(22\) |
risch | \(\frac {2 x^{2}-2 x -4}{\sqrt {-x^{2}+x +2}}+2 \arcsin \left (-\frac {1}{3}+\frac {2 x}{3}\right )\) | \(30\) |
trager | \(-2 \sqrt {-x^{2}+x +2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +\RootOf \left (\textit {\_Z}^{2}+1\right )+2 \sqrt {-x^{2}+x +2}\right )\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 21, normalized size = 0.78 \[ -2 \, \sqrt {-x^{2} + x + 2} - 2 \, \arcsin \left (-\frac {2}{3} \, x + \frac {1}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 40, normalized size = 1.48 \[ \mathrm {asin}\left (\frac {2\,x}{3}-\frac {1}{3}\right )-2\,\sqrt {-x^2+x+2}-\ln \left (x\,1{}\mathrm {i}+\sqrt {-x^2+x+2}-\frac {1}{2}{}\mathrm {i}\right )\,1{}\mathrm {i} \]
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 \[ \int \frac {2 x + 1}{\sqrt {- \left (x - 2\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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