Optimal. Leaf size=12 \[ 2 x \sqrt{x+e^x} \]
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Rubi [A] time = 0.258604, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {6742, 2273, 2262} \[ 2 x \sqrt{x+e^x} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2273
Rule 2262
Rubi steps
\begin{align*} \int \left (\frac{\left (1+e^x\right ) x}{\sqrt{e^x+x}}+2 \sqrt{e^x+x}\right ) \, dx &=2 \int \sqrt{e^x+x} \, dx+\int \frac{\left (1+e^x\right ) x}{\sqrt{e^x+x}} \, dx\\ &=2 \int \sqrt{e^x+x} \, dx+\int \left (\frac{x}{\sqrt{e^x+x}}+\frac{e^x x}{\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 = 0.0923953, size = 12, normalized size = 1. \[ 2 x \sqrt{x+e^x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, 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 = 1.13824, size = 22, normalized size = 1.83 \begin{align*} \frac{2 \,{\left (x^{2} + x e^{x}\right )}}{\sqrt{x + e^{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{\left (e^{x} + 1\right )}}{\sqrt{x + e^{x}}} + 2 \, \sqrt{x + e^{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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