Optimal. Leaf size=49 \[ \frac {5 x^2}{2}-\frac {50+51 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {7}{2} \log \left (1+x^2\right )-10 \log \left (2+x^2\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 49, 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^2}{2}-\frac {7}{2} \log \left (x^2+1\right )-10 \log \left (x^2+2\right )-\frac {51 x^2+50}{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^3 \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 \left (4+x+3 x^2+5 x^3\right )}{\left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {50+51 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \frac {24+12 x-5 x^2}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {50+51 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \left (-5+\frac {34+27 x}{2+3 x+x^2}\right ) \, dx,x,x^2\right )\\ &=\frac {5 x^2}{2}-\frac {50+51 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \text {Subst}\left (\int \frac {34+27 x}{2+3 x+x^2} \, dx,x,x^2\right )\\ &=\frac {5 x^2}{2}-\frac {50+51 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {7}{2} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^2\right )-10 \text {Subst}\left (\int \frac {1}{2+x} \, dx,x,x^2\right )\\ &=\frac {5 x^2}{2}-\frac {50+51 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {7}{2} \log \left (1+x^2\right )-10 \log \left (2+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 1.00 \begin {gather*} \frac {5 x^2}{2}+\frac {-50-51 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac {7}{2} \log \left (1+x^2\right )-10 \log \left (2+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 41, normalized size = 0.84
method | result | size |
default | \(\frac {5 x^{2}}{2}+\frac {1}{2 x^{2}+2}-\frac {7 \ln \left (x^{2}+1\right )}{2}-10 \ln \left (x^{2}+2\right )-\frac {26}{x^{2}+2}\) | \(41\) |
norman | \(\frac {-43 x^{2}+\frac {5}{2} x^{6}-40}{x^{4}+3 x^{2}+2}-\frac {7 \ln \left (x^{2}+1\right )}{2}-10 \ln \left (x^{2}+2\right )\) | \(43\) |
risch | \(\frac {5 x^{2}}{2}+\frac {-\frac {51 x^{2}}{2}-25}{x^{4}+3 x^{2}+2}-\frac {7 \ln \left (x^{2}+1\right )}{2}-10 \ln \left (x^{2}+2\right )\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 43, normalized size = 0.88 \begin {gather*} \frac {5}{2} \, x^{2} - \frac {51 \, x^{2} + 50}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - 10 \, \log \left (x^{2} + 2\right ) - \frac {7}{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 = 67, normalized size = 1.37 \begin {gather*} \frac {5 \, x^{6} + 15 \, x^{4} - 41 \, x^{2} - 20 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) - 7 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) - 50}{2 \, {\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 = 44, normalized size = 0.90 \begin {gather*} \frac {5 x^{2}}{2} + \frac {- 51 x^{2} - 50}{2 x^{4} + 6 x^{2} + 4} - \frac {7 \log {\left (x^{2} + 1 \right )}}{2} - 10 \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 = 2.65, size = 45, normalized size = 0.92 \begin {gather*} \frac {5}{2} \, x^{2} - \frac {51 \, x^{2} + 50}{2 \, {\left (x^{2} + 2\right )} {\left (x^{2} + 1\right )}} - 10 \, \log \left (x^{2} + 2\right ) - \frac {7}{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 = 0.04, size = 43, normalized size = 0.88 \begin {gather*} \frac {5\,x^2}{2}-10\,\ln \left (x^2+2\right )-\frac {\frac {51\,x^2}{2}+25}{x^4+3\,x^2+2}-\frac {7\,\ln \left (x^2+1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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