Optimal. Leaf size=41 \[ \frac {x^3}{6}+\frac {x^2}{2}+\frac {3}{4} \log \left (x^2-4 x+5\right )+\frac {3 x}{2}+6 \tan ^{-1}(2-x) \]
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Rubi [A] time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1657, 634, 618, 204, 628} \[ \frac {x^3}{6}+\frac {x^2}{2}+\frac {3}{4} \log \left (x^2-4 x+5\right )+\frac {3 x}{2}+6 \tan ^{-1}(2-x) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1657
Rubi steps
\begin {align*} \int \frac {-3+x-2 x^3+x^4}{10-8 x+2 x^2} \, dx &=\int \left (\frac {3}{2}+x+\frac {x^2}{2}-\frac {3 (6-x)}{10-8 x+2 x^2}\right ) \, dx\\ &=\frac {3 x}{2}+\frac {x^2}{2}+\frac {x^3}{6}-3 \int \frac {6-x}{10-8 x+2 x^2} \, dx\\ &=\frac {3 x}{2}+\frac {x^2}{2}+\frac {x^3}{6}+\frac {3}{4} \int \frac {-8+4 x}{10-8 x+2 x^2} \, dx-12 \int \frac {1}{10-8 x+2 x^2} \, dx\\ &=\frac {3 x}{2}+\frac {x^2}{2}+\frac {x^3}{6}+\frac {3}{4} \log \left (5-4 x+x^2\right )+24 \operatorname {Subst}\left (\int \frac {1}{-16-x^2} \, dx,x,-8+4 x\right )\\ &=\frac {3 x}{2}+\frac {x^2}{2}+\frac {x^3}{6}+6 \tan ^{-1}(2-x)+\frac {3}{4} \log \left (5-4 x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.95 \[ \frac {1}{2} \left (\frac {x^3}{3}+x^2+\frac {3}{2} \log \left (x^2-4 x+5\right )+3 x+12 \tan ^{-1}(2-x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 31, normalized size = 0.76 \[ \frac {1}{6} \, x^{3} + \frac {1}{2} \, x^{2} + \frac {3}{2} \, x - 6 \, \arctan \left (x - 2\right ) + \frac {3}{4} \, \log \left (x^{2} - 4 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 31, normalized size = 0.76 \[ \frac {1}{6} \, x^{3} + \frac {1}{2} \, x^{2} + \frac {3}{2} \, x - 6 \, \arctan \left (x - 2\right ) + \frac {3}{4} \, \log \left (x^{2} - 4 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.78 \[ \frac {x^{3}}{6}+\frac {x^{2}}{2}+\frac {3 x}{2}-6 \arctan \left (x -2\right )+\frac {3 \ln \left (x^{2}-4 x +5\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.07, size = 31, normalized size = 0.76 \[ \frac {1}{6} \, x^{3} + \frac {1}{2} \, x^{2} + \frac {3}{2} \, x - 6 \, \arctan \left (x - 2\right ) + \frac {3}{4} \, \log \left (x^{2} - 4 \, x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 31, normalized size = 0.76 \[ \frac {3\,x}{2}-6\,\mathrm {atan}\left (x-2\right )+\frac {3\,\ln \left (x^2-4\,x+5\right )}{4}+\frac {x^2}{2}+\frac {x^3}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 34, normalized size = 0.83 \[ \frac {x^{3}}{6} + \frac {x^{2}}{2} + \frac {3 x}{2} + \frac {3 \log {\left (x^{2} - 4 x + 5 \right )}}{4} - 6 \operatorname {atan}{\left (x - 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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