Optimal. Leaf size=47 \[ -\frac {x^2}{2}+x^3-\frac {\tan ^{-1}\left (\frac {1-4 x}{\sqrt {7}}\right )}{2 \sqrt {7}}+\frac {1}{4} \log \left (1-x+2 x^2\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1608, 1642,
648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-4 x}{\sqrt {7}}\right )}{2 \sqrt {7}}+x^3-\frac {x^2}{2}+\frac {1}{4} \log \left (2 x^2-x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1608
Rule 1642
Rubi steps
\begin {align*} \int \frac {4 x^2-5 x^3+6 x^4}{1-x+2 x^2} \, dx &=\int \frac {x^2 \left (4-5 x+6 x^2\right )}{1-x+2 x^2} \, dx\\ &=\int \left (-x+3 x^2+\frac {x}{1-x+2 x^2}\right ) \, dx\\ &=-\frac {x^2}{2}+x^3+\int \frac {x}{1-x+2 x^2} \, dx\\ &=-\frac {x^2}{2}+x^3+\frac {1}{4} \int \frac {1}{1-x+2 x^2} \, dx+\frac {1}{4} \int \frac {-1+4 x}{1-x+2 x^2} \, dx\\ &=-\frac {x^2}{2}+x^3+\frac {1}{4} \log \left (1-x+2 x^2\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,-1+4 x\right )\\ &=-\frac {x^2}{2}+x^3-\frac {\tan ^{-1}\left (\frac {1-4 x}{\sqrt {7}}\right )}{2 \sqrt {7}}+\frac {1}{4} \log \left (1-x+2 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 1.00 \begin {gather*} -\frac {x^2}{2}+x^3+\frac {\tan ^{-1}\left (\frac {-1+4 x}{\sqrt {7}}\right )}{2 \sqrt {7}}+\frac {1}{4} \log \left (1-x+2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 39, normalized size = 0.83
method | result | size |
default | \(x^{3}-\frac {x^{2}}{2}+\frac {\ln \left (2 x^{2}-x +1\right )}{4}+\frac {\sqrt {7}\, \arctan \left (\frac {\left (-1+4 x \right ) \sqrt {7}}{7}\right )}{14}\) | \(39\) |
risch | \(x^{3}-\frac {x^{2}}{2}+\frac {\ln \left (16 x^{2}-8 x +8\right )}{4}+\frac {\sqrt {7}\, \arctan \left (\frac {\left (-1+4 x \right ) \sqrt {7}}{7}\right )}{14}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.70, size = 38, normalized size = 0.81 \begin {gather*} x^{3} - \frac {1}{2} \, x^{2} + \frac {1}{14} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (4 \, x - 1\right )}\right ) + \frac {1}{4} \, \log \left (2 \, x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 38, normalized size = 0.81 \begin {gather*} x^{3} - \frac {1}{2} \, x^{2} + \frac {1}{14} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (4 \, x - 1\right )}\right ) + \frac {1}{4} \, \log \left (2 \, x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 46, normalized size = 0.98 \begin {gather*} x^{3} - \frac {x^{2}}{2} + \frac {\log {\left (x^{2} - \frac {x}{2} + \frac {1}{2} \right )}}{4} + \frac {\sqrt {7} \operatorname {atan}{\left (\frac {4 \sqrt {7} x}{7} - \frac {\sqrt {7}}{7} \right )}}{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 38, normalized size = 0.81 \begin {gather*} x^{3} - \frac {1}{2} \, x^{2} + \frac {1}{14} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (4 \, x - 1\right )}\right ) + \frac {1}{4} \, \log \left (2 \, x^{2} - x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 40, normalized size = 0.85 \begin {gather*} \frac {\ln \left (2\,x^2-x+1\right )}{4}+\frac {\sqrt {7}\,\mathrm {atan}\left (\frac {4\,\sqrt {7}\,x}{7}-\frac {\sqrt {7}}{7}\right )}{14}-\frac {x^2}{2}+x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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