Optimal. Leaf size=27 \[ -2 \sqrt {2+x-x^2}-2 \sin ^{-1}\left (\frac {1}{3} (1-2 x)\right ) \]
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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 = {654, 633, 222}
\begin {gather*} -2 \text {ArcSin}\left (\frac {1}{3} (1-2 x)\right )-2 \sqrt {-x^2+x+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 633
Rule 654
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} \text {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 = 0.09, size = 36, normalized size = 1.33 \begin {gather*} -2 \sqrt {2+x-x^2}-4 \tan ^{-1}\left (\frac {\sqrt {2+x-x^2}}{1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, 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 | \(-2 \sqrt {-x^{2}+x +2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {-x^{2}+x +2}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.73, size = 21, normalized size = 0.78 \begin {gather*} -2 \, \sqrt {-x^{2} + x + 2} - 2 \, \arcsin \left (-\frac {2}{3} \, x + \frac {1}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs.
\(2 (21) = 42\).
time = 0.62, size = 43, normalized size = 1.59 \begin {gather*} -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 ) \end {gather*}
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 \begin {gather*} \int \frac {2 x + 1}{\sqrt {- \left (x - 2\right ) \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.70, size = 21, normalized size = 0.78 \begin {gather*} -2 \, \sqrt {-x^{2} + x + 2} + 2 \, \arcsin \left (\frac {2}{3} \, x - \frac {1}{3}\right ) \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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