Optimal. Leaf size=74 \[ \frac {5 x^4}{4}-\frac {17 x^2}{2}+\frac {203 \tan ^{-1}\left (\frac {x^2+1}{\sqrt {2}}\right )}{8 \sqrt {2}}+\frac {25 \left (3-x^2\right )}{8 \left (x^4+2 x^2+3\right )}+\frac {19}{4} \log \left (x^4+2 x^2+3\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 74, 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^4}{4}-\frac {17 x^2}{2}+\frac {25 \left (3-x^2\right )}{8 \left (x^4+2 x^2+3\right )}+\frac {19}{4} \log \left (x^4+2 x^2+3\right )+\frac {203 \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^5 \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^2 \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 {150-56 x^2+40 x^3}{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 (-136+40 x+\frac {2 (279+76 x)}{3+2 x+x^2}\right ) \, dx,x,x^2\right )\\ &=-\frac {17 x^2}{2}+\frac {5 x^4}{4}+\frac {25 \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {279+76 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {17 x^2}{2}+\frac {5 x^4}{4}+\frac {25 \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {19}{4} \operatorname {Subst}\left (\int \frac {2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )+\frac {203}{8} \operatorname {Subst}\left (\int \frac {1}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {17 x^2}{2}+\frac {5 x^4}{4}+\frac {25 \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {19}{4} \log \left (3+2 x^2+x^4\right )-\frac {203}{4} \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=-\frac {17 x^2}{2}+\frac {5 x^4}{4}+\frac {25 \left (3-x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac {203 \tan ^{-1}\left (\frac {1+x^2}{\sqrt {2}}\right )}{8 \sqrt {2}}+\frac {19}{4} \log \left (3+2 x^2+x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 0.89 \begin {gather*} \frac {1}{16} \left (20 x^4-136 x^2+203 \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}+76 \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^5 \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 = 1.05, size = 85, normalized size = 1.15 \begin {gather*} \frac {20 \, x^{8} - 96 \, x^{6} - 212 \, x^{4} + 203 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - 458 \, x^{2} + 76 \, {\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 = 1.09, size = 66, normalized size = 0.89 \begin {gather*} \frac {5}{4} \, x^{4} - \frac {17}{2} \, x^{2} + \frac {203}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - \frac {38 \, x^{4} + 101 \, x^{2} + 39}{8 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac {19}{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 = 64, normalized size = 0.86 \begin {gather*} \frac {5 x^{4}}{4}-\frac {17 x^{2}}{2}+\frac {203 \sqrt {2}\, \arctan \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4}\right )}{16}+\frac {19 \ln \left (x^{4}+2 x^{2}+3\right )}{4}+\frac {-\frac {25 x^{2}}{4}+\frac {75}{4}}{2 x^{4}+4 x^{2}+6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.59, size = 59, normalized size = 0.80 \begin {gather*} \frac {5}{4} \, x^{4} - \frac {17}{2} \, x^{2} + \frac {203}{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 {19}{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 = 0.05, size = 65, normalized size = 0.88 \begin {gather*} \frac {19\,\ln \left (x^4+2\,x^2+3\right )}{4}-\frac {\frac {25\,x^2}{8}-\frac {75}{8}}{x^4+2\,x^2+3}+\frac {203\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x^2}{2}+\frac {\sqrt {2}}{2}\right )}{16}-\frac {17\,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 = 0.18, size = 73, normalized size = 0.99 \begin {gather*} \frac {5 x^{4}}{4} - \frac {17 x^{2}}{2} + \frac {75 - 25 x^{2}}{8 x^{4} + 16 x^{2} + 24} + \frac {19 \log {\left (x^{4} + 2 x^{2} + 3 \right )}}{4} + \frac {203 \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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