Optimal. Leaf size=13 \[ -\frac{1}{4 \left ((x+2)^2+1\right )^2} \]
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Rubi [A] time = 0.005537, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {372, 261} \[ -\frac{1}{4 \left ((x+2)^2+1\right )^2} \]
Antiderivative was successfully verified.
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Rule 372
Rule 261
Rubi steps
\begin{align*} \int \frac{2+x}{\left (1+(2+x)^2\right )^3} \, dx &=\operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right )^3} \, dx,x,2+x\right )\\ &=-\frac{1}{4 \left (1+(2+x)^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0029762, size = 13, normalized size = 1. \[ -\frac{1}{4 \left ((x+2)^2+1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 13, normalized size = 1. \begin{align*} -{\frac{1}{4\, \left ({x}^{2}+4\,x+5 \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0626, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{4 \,{\left ({\left (x + 2\right )}^{2} + 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44118, size = 55, normalized size = 4.23 \begin{align*} -\frac{1}{4 \,{\left (x^{4} + 8 \, x^{3} + 26 \, x^{2} + 40 \, x + 25\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.111223, size = 22, normalized size = 1.69 \begin{align*} - \frac{1}{4 x^{4} + 32 x^{3} + 104 x^{2} + 160 x + 100} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09604, size = 16, normalized size = 1.23 \begin{align*} -\frac{1}{4 \,{\left (x^{2} + 4 \, x + 5\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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