Optimal. Leaf size=30 \[ \frac {1}{3} \log \left (3 x^2+2\right )-\frac {5 \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {6}} \]
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Rubi [A] time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {635, 203, 260} \[ \frac {1}{3} \log \left (3 x^2+2\right )-\frac {5 \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {6}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rubi steps
\begin {align*} \int \frac {-5+2 x}{2+3 x^2} \, dx &=2 \int \frac {x}{2+3 x^2} \, dx-5 \int \frac {1}{2+3 x^2} \, dx\\ &=-\frac {5 \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {6}}+\frac {1}{3} \log \left (2+3 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ \frac {1}{3} \log \left (3 x^2+2\right )-\frac {5 \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {6}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 30, normalized size = 1.00 \[ \frac {1}{3} \log \left (3 x^2+2\right )-\frac {5 \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {6}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 23, normalized size = 0.77 \[ -\frac {5}{6} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {1}{3} \, \log \left (3 \, x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.85, size = 21, normalized size = 0.70 \[ -\frac {5}{6} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {1}{3} \, \log \left (x^{2} + \frac {2}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 24, normalized size = 0.80
method | result | size |
default | \(\frac {\ln \left (3 x^{2}+2\right )}{3}-\frac {5 \arctan \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {6}}{6}\) | \(24\) |
risch | \(\frac {\ln \left (9 x^{2}+6\right )}{3}-\frac {5 \arctan \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {6}}{6}\) | \(24\) |
meijerg | \(-\frac {5 \sqrt {6}\, \arctan \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{6}+\frac {\ln \left (1+\frac {3 x^{2}}{2}\right )}{3}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 23, normalized size = 0.77 \[ -\frac {5}{6} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {1}{3} \, \log \left (3 \, x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 21, normalized size = 0.70 \[ \frac {\ln \left (x^2+\frac {2}{3}\right )}{3}-\frac {5\,\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x}{2}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 27, normalized size = 0.90 \[ \frac {\log {\left (x^{2} + \frac {2}{3} \right )}}{3} - \frac {5 \sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} x}{2} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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