Optimal. Leaf size=86 \[ \frac {5 x^8}{8}-\frac {17 x^6}{6}+\frac {19 x^4}{4}+19 x^2+\frac {201 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{8 \sqrt {2}}-\frac {25 \left (7 x^2+15\right )}{8 \left (x^4+2 x^2+3\right )}-\frac {183}{4} \log \left (x^4+2 x^2+3\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1663, 1660, 1657, 634, 618, 204, 628} \[ \frac {5 x^8}{8}-\frac {17 x^6}{6}+\frac {19 x^4}{4}+19 x^2-\frac {25 \left (7 x^2+15\right )}{8 \left (x^4+2 x^2+3\right )}-\frac {183}{4} \log \left (x^4+2 x^2+3\right )+\frac {201 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{8 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1657
Rule 1660
Rule 1663
Rubi steps
\begin {align*} \int \frac {x^9 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^4 \left (4+x+3 x^2+5 x^3\right )}{\left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {25 \left (15+7 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \operatorname {Subst}\left (\int \frac {-150-400 x+200 x^2-56 x^4+40 x^5}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {25 \left (15+7 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \operatorname {Subst}\left (\int \left (304+152 x-136 x^2+40 x^3-\frac {6 (177+244 x)}{3+2 x+x^2}\right ) \, dx,x,x^2\right )\\ &=19 x^2+\frac {19 x^4}{4}-\frac {17 x^6}{6}+\frac {5 x^8}{8}-\frac {25 \left (15+7 x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {3}{8} \operatorname {Subst}\left (\int \frac {177+244 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=19 x^2+\frac {19 x^4}{4}-\frac {17 x^6}{6}+\frac {5 x^8}{8}-\frac {25 \left (15+7 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {201}{8} \operatorname {Subst}\left (\int \frac {1}{3+2 x+x^2} \, dx,x,x^2\right )-\frac {183}{4} \operatorname {Subst}\left (\int \frac {2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=19 x^2+\frac {19 x^4}{4}-\frac {17 x^6}{6}+\frac {5 x^8}{8}-\frac {25 \left (15+7 x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {183}{4} \log \left (3+2 x^2+x^4\right )-\frac {201}{4} \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=19 x^2+\frac {19 x^4}{4}-\frac {17 x^6}{6}+\frac {5 x^8}{8}-\frac {25 \left (15+7 x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {201 \tan ^{-1}\left (\frac {1+x^2}{\sqrt {2}}\right )}{8 \sqrt {2}}-\frac {183}{4} \log \left (3+2 x^2+x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 78, normalized size = 0.91 \[ \frac {1}{48} \left (30 x^8-136 x^6+228 x^4+912 x^2+603 \sqrt {2} \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )-\frac {150 \left (7 x^2+15\right )}{x^4+2 x^2+3}-2196 \log \left (x^4+2 x^2+3\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 95, normalized size = 1.10 \[ \frac {30 \, x^{12} - 76 \, x^{10} + 46 \, x^{8} + 960 \, x^{6} + 2508 \, x^{4} + 603 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) + 1686 \, x^{2} - 2196 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 2250}{48 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 76, normalized size = 0.88 \[ \frac {5}{8} \, x^{8} - \frac {17}{6} \, x^{6} + \frac {19}{4} \, x^{4} + 19 \, x^{2} + \frac {201}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) + \frac {366 \, x^{4} + 557 \, x^{2} + 723}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {183}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.86 \[ \frac {5 x^{8}}{8}-\frac {17 x^{6}}{6}+\frac {19 x^{4}}{4}+19 x^{2}+\frac {201 \sqrt {2}\, \arctan \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4}\right )}{16}-\frac {183 \ln \left (x^{4}+2 x^{2}+3\right )}{4}-\frac {\frac {175 x^{2}}{4}+\frac {375}{4}}{2 \left (x^{4}+2 x^{2}+3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 71, normalized size = 0.83 \[ \frac {5}{8} \, x^{8} - \frac {17}{6} \, x^{6} + \frac {19}{4} \, x^{4} + 19 \, x^{2} + \frac {201}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - \frac {25 \, {\left (7 \, x^{2} + 15\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {183}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 75, normalized size = 0.87 \[ \frac {201\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x^2}{2}+\frac {\sqrt {2}}{2}\right )}{16}-\frac {\frac {175\,x^2}{8}+\frac {375}{8}}{x^4+2\,x^2+3}-\frac {183\,\ln \left (x^4+2\,x^2+3\right )}{4}+19\,x^2+\frac {19\,x^4}{4}-\frac {17\,x^6}{6}+\frac {5\,x^8}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 87, normalized size = 1.01 \[ \frac {5 x^{8}}{8} - \frac {17 x^{6}}{6} + \frac {19 x^{4}}{4} + 19 x^{2} + \frac {- 175 x^{2} - 375}{8 x^{4} + 16 x^{2} + 24} - \frac {183 \log {\left (x^{4} + 2 x^{2} + 3 \right )}}{4} + \frac {201 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x^{2}}{2} + \frac {\sqrt {2}}{2} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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