Optimal. Leaf size=46 \[ \frac {5}{8} \log \left (x^2+x+2\right )-\frac {1}{2 x}-\frac {\log (x)}{4}+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {7}}\right )}{4 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1594, 1628, 634, 618, 204, 628} \begin {gather*} \frac {5}{8} \log \left (x^2+x+2\right )-\frac {1}{2 x}-\frac {\log (x)}{4}+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {7}}\right )}{4 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1594
Rule 1628
Rubi steps
\begin {align*} \int \frac {1+x^2+x^3}{2 x^2+x^3+x^4} \, dx &=\int \frac {1+x^2+x^3}{x^2 \left (2+x+x^2\right )} \, dx\\ &=\int \left (\frac {1}{2 x^2}-\frac {1}{4 x}+\frac {3+5 x}{4 \left (2+x+x^2\right )}\right ) \, dx\\ &=-\frac {1}{2 x}-\frac {\log (x)}{4}+\frac {1}{4} \int \frac {3+5 x}{2+x+x^2} \, dx\\ &=-\frac {1}{2 x}-\frac {\log (x)}{4}+\frac {1}{8} \int \frac {1}{2+x+x^2} \, dx+\frac {5}{8} \int \frac {1+2 x}{2+x+x^2} \, dx\\ &=-\frac {1}{2 x}-\frac {\log (x)}{4}+\frac {5}{8} \log \left (2+x+x^2\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,1+2 x\right )\\ &=-\frac {1}{2 x}+\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {7}}\right )}{4 \sqrt {7}}-\frac {\log (x)}{4}+\frac {5}{8} \log \left (2+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 1.00 \begin {gather*} \frac {5}{8} \log \left (x^2+x+2\right )-\frac {1}{2 x}-\frac {\log (x)}{4}+\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {7}}\right )}{4 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2+x^3}{2 x^2+x^3+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.52, size = 39, normalized size = 0.85 \begin {gather*} \frac {2 \, \sqrt {7} x \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x + 1\right )}\right ) + 35 \, x \log \left (x^{2} + x + 2\right ) - 14 \, x \log \relax (x) - 28}{56 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 36, normalized size = 0.78 \begin {gather*} \frac {1}{28} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x + 1\right )}\right ) - \frac {1}{2 \, x} + \frac {5}{8} \, \log \left (x^{2} + x + 2\right ) - \frac {1}{4} \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.78 \begin {gather*} \frac {\sqrt {7}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {7}}{7}\right )}{28}-\frac {\ln \relax (x )}{4}+\frac {5 \ln \left (x^{2}+x +2\right )}{8}-\frac {1}{2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.09, size = 35, normalized size = 0.76 \begin {gather*} \frac {1}{28} \, \sqrt {7} \arctan \left (\frac {1}{7} \, \sqrt {7} {\left (2 \, x + 1\right )}\right ) - \frac {1}{2 \, x} + \frac {5}{8} \, \log \left (x^{2} + x + 2\right ) - \frac {1}{4} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.17, size = 49, normalized size = 1.07 \begin {gather*} -\frac {\ln \relax (x)}{4}-\ln \left (x+\frac {1}{2}-\frac {\sqrt {7}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {5}{8}+\frac {\sqrt {7}\,1{}\mathrm {i}}{56}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {7}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {5}{8}+\frac {\sqrt {7}\,1{}\mathrm {i}}{56}\right )-\frac {1}{2\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 46, normalized size = 1.00 \begin {gather*} - \frac {\log {\relax (x )}}{4} + \frac {5 \log {\left (x^{2} + x + 2 \right )}}{8} + \frac {\sqrt {7} \operatorname {atan}{\left (\frac {2 \sqrt {7} x}{7} + \frac {\sqrt {7}}{7} \right )}}{28} - \frac {1}{2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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