Optimal. Leaf size=25 \[ -\log \left (x^2+2 x+5\right )+x-\frac{3}{2} \tan ^{-1}\left (\frac{x+1}{2}\right ) \]
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Rubi [A] time = 0.0159191, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {703, 634, 618, 204, 628} \[ -\log \left (x^2+2 x+5\right )+x-\frac{3}{2} \tan ^{-1}\left (\frac{x+1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 703
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^2}{5+2 x+x^2} \, dx &=x+\int \frac{-5-2 x}{5+2 x+x^2} \, dx\\ &=x-3 \int \frac{1}{5+2 x+x^2} \, dx-\int \frac{2+2 x}{5+2 x+x^2} \, dx\\ &=x-\log \left (5+2 x+x^2\right )+6 \operatorname{Subst}\left (\int \frac{1}{-16-x^2} \, dx,x,2+2 x\right )\\ &=x-\frac{3}{2} \tan ^{-1}\left (\frac{1+x}{2}\right )-\log \left (5+2 x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0036703, size = 25, normalized size = 1. \[ -\log \left (x^2+2 x+5\right )+x-\frac{3}{2} \tan ^{-1}\left (\frac{x+1}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 22, normalized size = 0.9 \begin{align*} x-{\frac{3}{2}\arctan \left ({\frac{1}{2}}+{\frac{x}{2}} \right ) }-\ln \left ({x}^{2}+2\,x+5 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40652, size = 28, normalized size = 1.12 \begin{align*} x - \frac{3}{2} \, \arctan \left (\frac{1}{2} \, x + \frac{1}{2}\right ) - \log \left (x^{2} + 2 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75202, size = 68, normalized size = 2.72 \begin{align*} x - \frac{3}{2} \, \arctan \left (\frac{1}{2} \, x + \frac{1}{2}\right ) - \log \left (x^{2} + 2 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.096622, size = 22, normalized size = 0.88 \begin{align*} x - \log{\left (x^{2} + 2 x + 5 \right )} - \frac{3 \operatorname{atan}{\left (\frac{x}{2} + \frac{1}{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04478, size = 28, normalized size = 1.12 \begin{align*} x - \frac{3}{2} \, \arctan \left (\frac{1}{2} \, x + \frac{1}{2}\right ) - \log \left (x^{2} + 2 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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