Optimal. Leaf size=12 \[ 2 x \sqrt{x+e^x} \]
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Rubi [A] time = 0.130266, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {2273, 2262} \[ 2 x \sqrt{x+e^x} \]
Antiderivative was successfully verified.
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Rule 2273
Rule 2262
Rubi steps
\begin{align*} \int \left (\frac{x}{\sqrt{e^x+x}}+\frac{e^x x}{\sqrt{e^x+x}}+2 \sqrt{e^x+x}\right ) \, dx &=2 \int \sqrt{e^x+x} \, dx+\int \frac{x}{\sqrt{e^x+x}} \, dx+\int \frac{e^x x}{\sqrt{e^x+x}} \, dx\\ &=-2 \sqrt{e^x+x}+2 x \sqrt{e^x+x}+\int \frac{1}{\sqrt{e^x+x}} \, dx-\int \frac{x}{\sqrt{e^x+x}} \, dx+\int \sqrt{e^x+x} \, dx\\ &=2 x \sqrt{e^x+x}\\ \end{align*}
Mathematica [A] time = 0.0446522, size = 12, normalized size = 1. \[ 2 x \sqrt{x+e^x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 10, normalized size = 0.8 \begin{align*} 2\,x\sqrt{{{\rm e}^{x}}+x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08326, size = 12, normalized size = 1. \begin{align*} 2 \, \sqrt{x + e^{x}} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x e^{x} + 3 x + 2 e^{x}}{\sqrt{x + e^{x}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x e^{x}}{\sqrt{x + e^{x}}} + 2 \, \sqrt{x + e^{x}} + \frac{x}{\sqrt{x + e^{x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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