Optimal. Leaf size=54 \[ \frac {5 x^4}{4}-\frac {27 x^2}{2}+3 \log \left (x^2+1\right )+46 \log \left (x^2+2\right )+\frac {103 x^2+102}{2 \left (x^4+3 x^2+2\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1663, 1660, 1657, 632, 31} \begin {gather*} \frac {5 x^4}{4}-\frac {27 x^2}{2}+\frac {103 x^2+102}{2 \left (x^4+3 x^2+2\right )}+3 \log \left (x^2+1\right )+46 \log \left (x^2+2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 632
Rule 1657
Rule 1660
Rule 1663
Rubi steps
\begin {align*} \int \frac {x^5 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 \left (4+x+3 x^2+5 x^3\right )}{\left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {102+103 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {-50-27 x+12 x^2-5 x^3}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=\frac {102+103 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \operatorname {Subst}\left (\int \left (27-5 x-\frac {2 (52+49 x)}{2+3 x+x^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {27 x^2}{2}+\frac {5 x^4}{4}+\frac {102+103 x^2}{2 \left (2+3 x^2+x^4\right )}+\operatorname {Subst}\left (\int \frac {52+49 x}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {27 x^2}{2}+\frac {5 x^4}{4}+\frac {102+103 x^2}{2 \left (2+3 x^2+x^4\right )}+3 \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )+46 \operatorname {Subst}\left (\int \frac {1}{2+x} \, dx,x,x^2\right )\\ &=-\frac {27 x^2}{2}+\frac {5 x^4}{4}+\frac {102+103 x^2}{2 \left (2+3 x^2+x^4\right )}+3 \log \left (1+x^2\right )+46 \log \left (2+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 1.00 \begin {gather*} \frac {5 x^4}{4}-\frac {27 x^2}{2}+3 \log \left (x^2+1\right )+46 \log \left (x^2+2\right )+\frac {103 x^2+102}{2 \left (x^4+3 x^2+2\right )} \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 {x^5 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.22, size = 72, normalized size = 1.33 \begin {gather*} \frac {5 \, x^{8} - 39 \, x^{6} - 152 \, x^{4} + 98 \, x^{2} + 184 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 12 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) + 204}{4 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 53, normalized size = 0.98 \begin {gather*} \frac {5}{4} \, x^{4} - \frac {27}{2} \, x^{2} - \frac {49 \, x^{4} + 44 \, x^{2} - 4}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + 46 \, \log \left (x^{2} + 2\right ) + 3 \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 46, normalized size = 0.85 \begin {gather*} \frac {5 x^{4}}{4}-\frac {27 x^{2}}{2}+3 \ln \left (x^{2}+1\right )+46 \ln \left (x^{2}+2\right )-\frac {1}{2 \left (x^{2}+1\right )}+\frac {52}{x^{2}+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 48, normalized size = 0.89 \begin {gather*} \frac {5}{4} \, x^{4} - \frac {27}{2} \, x^{2} + \frac {103 \, x^{2} + 102}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + 46 \, \log \left (x^{2} + 2\right ) + 3 \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 47, normalized size = 0.87 \begin {gather*} 3\,\ln \left (x^2+1\right )+46\,\ln \left (x^2+2\right )+\frac {\frac {103\,x^2}{2}+51}{x^4+3\,x^2+2}-\frac {27\,x^2}{2}+\frac {5\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 48, normalized size = 0.89 \begin {gather*} \frac {5 x^{4}}{4} - \frac {27 x^{2}}{2} + \frac {103 x^{2} + 102}{2 x^{4} + 6 x^{2} + 4} + 3 \log {\left (x^{2} + 1 \right )} + 46 \log {\left (x^{2} + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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