Optimal. Leaf size=68 \[ -\frac {293 x^2}{2}+\frac {49 x^4}{2}-\frac {9 x^6}{2}+\frac {5 x^8}{8}+\frac {414+415 x^2}{2 \left (2+3 x^2+x^4\right )}+2 \log \left (1+x^2\right )+392 \log \left (2+x^2\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 68, 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 = {1677, 1674,
1671, 646, 31} \begin {gather*} \frac {5 x^8}{8}-\frac {9 x^6}{2}+\frac {49 x^4}{2}-\frac {293 x^2}{2}+2 \log \left (x^2+1\right )+392 \log \left (x^2+2\right )+\frac {415 x^2+414}{2 \left (x^4+3 x^2+2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 646
Rule 1671
Rule 1674
Rule 1677
Rubi steps
\begin {align*} \int \frac {x^9 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4 \left (4+x+3 x^2+5 x^3\right )}{\left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {414+415 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \frac {-206-105 x+53 x^2-27 x^3+12 x^4-5 x^5}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=\frac {414+415 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \left (293-98 x+27 x^2-5 x^3-\frac {4 (198+197 x)}{2+3 x+x^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {293 x^2}{2}+\frac {49 x^4}{2}-\frac {9 x^6}{2}+\frac {5 x^8}{8}+\frac {414+415 x^2}{2 \left (2+3 x^2+x^4\right )}+2 \text {Subst}\left (\int \frac {198+197 x}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {293 x^2}{2}+\frac {49 x^4}{2}-\frac {9 x^6}{2}+\frac {5 x^8}{8}+\frac {414+415 x^2}{2 \left (2+3 x^2+x^4\right )}+2 \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )+392 \text {Subst}\left (\int \frac {1}{2+x} \, dx,x,x^2\right )\\ &=-\frac {293 x^2}{2}+\frac {49 x^4}{2}-\frac {9 x^6}{2}+\frac {5 x^8}{8}+\frac {414+415 x^2}{2 \left (2+3 x^2+x^4\right )}+2 \log \left (1+x^2\right )+392 \log \left (2+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 62, normalized size = 0.91 \begin {gather*} \frac {1}{8} \left (-1172 x^2+196 x^4-36 x^6+5 x^8+\frac {4 \left (414+415 x^2\right )}{2+3 x^2+x^4}+16 \log \left (1+x^2\right )+3136 \log \left (2+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 56, normalized size = 0.82
method | result | size |
default | \(\frac {5 x^{8}}{8}-\frac {9 x^{6}}{2}+\frac {49 x^{4}}{2}-\frac {293 x^{2}}{2}-\frac {1}{2 \left (x^{2}+1\right )}+2 \ln \left (x^{2}+1\right )+392 \ln \left (x^{2}+2\right )+\frac {208}{x^{2}+2}\) | \(56\) |
norman | \(\frac {1086 x^{2}-82 x^{6}+\frac {49}{4} x^{8}-\frac {21}{8} x^{10}+\frac {5}{8} x^{12}+988}{x^{4}+3 x^{2}+2}+2 \ln \left (x^{2}+1\right )+392 \ln \left (x^{2}+2\right )\) | \(58\) |
risch | \(\frac {5 x^{8}}{8}-\frac {9 x^{6}}{2}+\frac {49 x^{4}}{2}-\frac {293 x^{2}}{2}+\frac {\frac {415 x^{2}}{2}+207}{x^{4}+3 x^{2}+2}+2 \ln \left (x^{2}+1\right )+392 \ln \left (x^{2}+2\right )\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 58, normalized size = 0.85 \begin {gather*} \frac {5}{8} \, x^{8} - \frac {9}{2} \, x^{6} + \frac {49}{2} \, x^{4} - \frac {293}{2} \, x^{2} + \frac {415 \, x^{2} + 414}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + 392 \, \log \left (x^{2} + 2\right ) + 2 \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 82, normalized size = 1.21 \begin {gather*} \frac {5 \, x^{12} - 21 \, x^{10} + 98 \, x^{8} - 656 \, x^{6} - 3124 \, x^{4} - 684 \, x^{2} + 3136 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 16 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) + 1656}{8 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 61, normalized size = 0.90 \begin {gather*} \frac {5 x^{8}}{8} - \frac {9 x^{6}}{2} + \frac {49 x^{4}}{2} - \frac {293 x^{2}}{2} + \frac {415 x^{2} + 414}{2 x^{4} + 6 x^{2} + 4} + 2 \log {\left (x^{2} + 1 \right )} + 392 \log {\left (x^{2} + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.83, size = 63, normalized size = 0.93 \begin {gather*} \frac {5}{8} \, x^{8} - \frac {9}{2} \, x^{6} + \frac {49}{2} \, x^{4} - \frac {293}{2} \, x^{2} - \frac {394 \, x^{4} + 767 \, x^{2} + 374}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} + 392 \, \log \left (x^{2} + 2\right ) + 2 \, \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 = 0.06, size = 57, normalized size = 0.84 \begin {gather*} 2\,\ln \left (x^2+1\right )+392\,\ln \left (x^2+2\right )+\frac {\frac {415\,x^2}{2}+207}{x^4+3\,x^2+2}-\frac {293\,x^2}{2}+\frac {49\,x^4}{2}-\frac {9\,x^6}{2}+\frac {5\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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