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 = 0.0147971, antiderivative size = 29, normalized size of antiderivative = 1., 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} \[ \frac{x^3}{3}-2 x^2+12 x-\frac{16}{x+2}-32 \log (x+2) \]
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*}
Mathematica [A] time = 0.009439, size = 30, normalized size = 1.03 \[ \frac{1}{3} \left (x^3-6 x^2+36 x-\frac{48}{x+2}-96 \log (x+2)+104\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 28, normalized size = 1. \begin{align*} 12\,x-2\,{x}^{2}+{\frac{{x}^{3}}{3}}-16\, \left ( 2+x \right ) ^{-1}-32\,\ln \left ( 2+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985296, size = 36, normalized size = 1.24 \begin{align*} \frac{1}{3} \, x^{3} - 2 \, x^{2} + 12 \, x - \frac{16}{x + 2} - 32 \, \log \left (x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2301, size = 97, normalized size = 3.34 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.08555, size = 24, normalized size = 0.83 \begin{align*} \frac{x^{3}}{3} - 2 x^{2} + 12 x - 32 \log{\left (x + 2 \right )} - \frac{16}{x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11598, size = 38, normalized size = 1.31 \begin{align*} \frac{1}{3} \, x^{3} - 2 \, x^{2} + 12 \, x - \frac{16}{x + 2} - 32 \, \log \left ({\left | x + 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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