Optimal. Leaf size=65 \[ \frac {5 x^2}{2}-\frac {17 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{8 \sqrt {2}}-\frac {25 \left (x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}-\frac {17}{4} \log \left (x^4+2 x^2+3\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 65, 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} \begin {gather*} \frac {5 x^2}{2}-\frac {25 \left (x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}-\frac {17}{4} \log \left (x^4+2 x^2+3\right )-\frac {17 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{8 \sqrt {2}} \end {gather*}
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^3 \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 \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 (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \operatorname {Subst}\left (\int \frac {-50-56 x+40 x^2}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {25 \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \operatorname {Subst}\left (\int \left (40-\frac {34 (5+4 x)}{3+2 x+x^2}\right ) \, dx,x,x^2\right )\\ &=\frac {5 x^2}{2}-\frac {25 \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {17}{8} \operatorname {Subst}\left (\int \frac {5+4 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac {5 x^2}{2}-\frac {25 \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {17}{8} \operatorname {Subst}\left (\int \frac {1}{3+2 x+x^2} \, dx,x,x^2\right )-\frac {17}{4} \operatorname {Subst}\left (\int \frac {2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac {5 x^2}{2}-\frac {25 \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {17}{4} \log \left (3+2 x^2+x^4\right )+\frac {17}{4} \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=\frac {5 x^2}{2}-\frac {25 \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac {17 \tan ^{-1}\left (\frac {1+x^2}{\sqrt {2}}\right )}{8 \sqrt {2}}-\frac {17}{4} \log \left (3+2 x^2+x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 61, normalized size = 0.94 \begin {gather*} \frac {1}{16} \left (40 x^2-17 \sqrt {2} \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )-\frac {50 \left (x^2+3\right )}{x^4+2 x^2+3}-68 \log \left (x^4+2 x^2+3\right )\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^3 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.00, size = 80, normalized size = 1.23 \begin {gather*} \frac {40 \, x^{6} + 80 \, x^{4} - 17 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) + 70 \, x^{2} - 68 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 150}{16 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 54, normalized size = 0.83 \begin {gather*} \frac {5}{2} \, x^{2} - \frac {17}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - \frac {25 \, {\left (x^{2} + 3\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {17}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 0.91 \begin {gather*} \frac {5 x^{2}}{2}-\frac {17 \sqrt {2}\, \arctan \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4}\right )}{16}-\frac {17 \ln \left (x^{4}+2 x^{2}+3\right )}{4}-\frac {\frac {25 x^{2}}{4}+\frac {75}{4}}{2 \left (x^{4}+2 x^{2}+3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.72, size = 54, normalized size = 0.83 \begin {gather*} \frac {5}{2} \, x^{2} - \frac {17}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - \frac {25 \, {\left (x^{2} + 3\right )}}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {17}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 60, normalized size = 0.92 \begin {gather*} \frac {5\,x^2}{2}-\frac {\frac {25\,x^2}{8}+\frac {75}{8}}{x^4+2\,x^2+3}-\frac {17\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x^2}{2}+\frac {\sqrt {2}}{2}\right )}{16}-\frac {17\,\ln \left (x^4+2\,x^2+3\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 68, normalized size = 1.05 \begin {gather*} \frac {5 x^{2}}{2} + \frac {- 25 x^{2} - 75}{8 x^{4} + 16 x^{2} + 24} - \frac {17 \log {\left (x^{4} + 2 x^{2} + 3 \right )}}{4} - \frac {17 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x^{2}}{2} + \frac {\sqrt {2}}{2} \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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