Optimal. Leaf size=29 \[ e^2+\frac {2}{3} \left (2+\frac {x}{x+\left (\frac {2}{x}+x\right )^2}\right )-\log (25) \]
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Rubi [A] time = 0.28, antiderivative size = 22, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 4, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1594, 6688, 12, 1588} \begin {gather*} \frac {2 x^3}{3 \left (x^4+x^3+4 x^2+4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1588
Rule 1594
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (24+8 x^2-2 x^4\right )}{48+96 x^2+24 x^3+72 x^4+24 x^5+27 x^6+6 x^7+3 x^8} \, dx\\ &=\int \frac {2 x^2 \left (12+4 x^2-x^4\right )}{3 \left (4+4 x^2+x^3+x^4\right )^2} \, dx\\ &=\frac {2}{3} \int \frac {x^2 \left (12+4 x^2-x^4\right )}{\left (4+4 x^2+x^3+x^4\right )^2} \, dx\\ &=\frac {2 x^3}{3 \left (4+4 x^2+x^3+x^4\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.76 \begin {gather*} \frac {2 x^3}{3 \left (4+4 x^2+x^3+x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 20, normalized size = 0.69 \begin {gather*} \frac {2 \, x^{3}}{3 \, {\left (x^{4} + x^{3} + 4 \, x^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 20, normalized size = 0.69 \begin {gather*} \frac {2 \, x^{3}}{3 \, {\left (x^{4} + x^{3} + 4 \, x^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 21, normalized size = 0.72
method | result | size |
gosper | \(\frac {2 x^{3}}{3 \left (x^{4}+x^{3}+4 x^{2}+4\right )}\) | \(21\) |
default | \(\frac {2 x^{3}}{3 \left (x^{4}+x^{3}+4 x^{2}+4\right )}\) | \(21\) |
norman | \(\frac {2 x^{3}}{3 \left (x^{4}+x^{3}+4 x^{2}+4\right )}\) | \(21\) |
risch | \(\frac {2 x^{3}}{3 \left (x^{4}+x^{3}+4 x^{2}+4\right )}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 20, normalized size = 0.69 \begin {gather*} \frac {2 \, x^{3}}{3 \, {\left (x^{4} + x^{3} + 4 \, x^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 24, normalized size = 0.83 \begin {gather*} \frac {2\,x^3}{3\,\left (x^4+x^3+4\,x^2+4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 20, normalized size = 0.69 \begin {gather*} \frac {2 x^{3}}{3 x^{4} + 3 x^{3} + 12 x^{2} + 12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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