Optimal. Leaf size=29 \[ \frac {x^3}{3}-2 x^2+12 x-\frac {16}{x+2}-32 \log (x+2) \]
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Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 43} \begin {gather*} \frac {x^3}{3}-2 x^2+12 x-\frac {16}{x+2}-32 \log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {x^4}{4+4 x+x^2} \, dx &=\int \frac {x^4}{(2+x)^2} \, dx\\ &=\int \left (12-4 x+x^2+\frac {16}{(2+x)^2}-\frac {32}{2+x}\right ) \, dx\\ &=12 x-2 x^2+\frac {x^3}{3}-\frac {16}{2+x}-32 \log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.03 \begin {gather*} \frac {1}{3} \left (x^3-6 x^2+36 x-\frac {48}{x+2}-96 \log (x+2)+104\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^4}{4+4 x+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 34, normalized size = 1.17 \begin {gather*} \frac {x^{4} - 4 \, x^{3} + 24 \, x^{2} - 96 \, {\left (x + 2\right )} \log \left (x + 2\right ) + 72 \, x - 48}{3 \, {\left (x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 28, normalized size = 0.97 \begin {gather*} \frac {1}{3} \, x^{3} - 2 \, x^{2} + 12 \, x - \frac {16}{x + 2} - 32 \, \log \left ({\left | x + 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 28, normalized size = 0.97 \begin {gather*} \frac {x^{3}}{3}-2 x^{2}+12 x -32 \ln \left (x +2\right )-\frac {16}{x +2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 27, normalized size = 0.93 \begin {gather*} \frac {1}{3} \, x^{3} - 2 \, x^{2} + 12 \, x - \frac {16}{x + 2} - 32 \, \log \left (x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 27, normalized size = 0.93 \begin {gather*} 12\,x-32\,\ln \left (x+2\right )-\frac {16}{x+2}-2\,x^2+\frac {x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 24, normalized size = 0.83 \begin {gather*} \frac {x^{3}}{3} - 2 x^{2} + 12 x - 32 \log {\left (x + 2 \right )} - \frac {16}{x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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