Optimal. Leaf size=40 \[ -\frac {1}{3} \sqrt {2+4 x-3 x^2}-\frac {2 \sin ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{3 \sqrt {3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {654, 633, 222}
\begin {gather*} -\frac {2 \text {ArcSin}\left (\frac {2-3 x}{\sqrt {10}}\right )}{3 \sqrt {3}}-\frac {1}{3} \sqrt {-3 x^2+4 x+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 633
Rule 654
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {2+4 x-3 x^2}} \, dx &=-\frac {1}{3} \sqrt {2+4 x-3 x^2}+\frac {2}{3} \int \frac {1}{\sqrt {2+4 x-3 x^2}} \, dx\\ &=-\frac {1}{3} \sqrt {2+4 x-3 x^2}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{40}}} \, dx,x,4-6 x\right )}{3 \sqrt {30}}\\ &=-\frac {1}{3} \sqrt {2+4 x-3 x^2}-\frac {2 \sin ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 60, normalized size = 1.50 \begin {gather*} -\frac {1}{3} \sqrt {2+4 x-3 x^2}-\frac {4 \tan ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {2}-\sqrt {2+4 x-3 x^2}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.85, size = 30, normalized size = 0.75
method | result | size |
default | \(-\frac {\sqrt {-3 x^{2}+4 x +2}}{3}+\frac {2 \sqrt {3}\, \arcsin \left (\frac {3 \sqrt {10}\, \left (x -\frac {2}{3}\right )}{10}\right )}{9}\) | \(30\) |
risch | \(\frac {3 x^{2}-4 x -2}{3 \sqrt {-3 x^{2}+4 x +2}}+\frac {2 \sqrt {3}\, \arcsin \left (\frac {3 \sqrt {10}\, \left (x -\frac {2}{3}\right )}{10}\right )}{9}\) | \(40\) |
trager | \(-\frac {\sqrt {-3 x^{2}+4 x +2}}{3}-\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (3 x \RootOf \left (\textit {\_Z}^{2}+3\right )-2 \RootOf \left (\textit {\_Z}^{2}+3\right )+3 \sqrt {-3 x^{2}+4 x +2}\right )}{9}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 31, normalized size = 0.78 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \arcsin \left (-\frac {1}{10} \, \sqrt {10} {\left (3 \, x - 2\right )}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.63, size = 55, normalized size = 1.38 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x + 2} {\left (3 \, x - 2\right )}}{3 \, {\left (3 \, x^{2} - 4 \, x - 2\right )}}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} \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 {x}{\sqrt {- 3 x^{2} + 4 x + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 31, normalized size = 0.78 \begin {gather*} \frac {2}{9} \, \sqrt {3} \arcsin \left (\frac {1}{10} \, \sqrt {10} {\left (3 \, x - 2\right )}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 46, normalized size = 1.15 \begin {gather*} -\frac {\sqrt {-3\,x^2+4\,x+2}}{3}-\frac {\sqrt {3}\,\ln \left (\sqrt {-3\,x^2+4\,x+2}+\frac {\sqrt {3}\,\left (3\,x-2\right )\,1{}\mathrm {i}}{3}\right )\,2{}\mathrm {i}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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