Optimal. Leaf size=26 \[ 2 x \sqrt{x+e^x}-2 \text{CannotIntegrate}\left (\sqrt{x+e^x},x\right ) \]
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Rubi [A] time = 0.18311, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (1+e^x\right ) x}{\sqrt{e^x+x}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (1+e^x\right ) x}{\sqrt{e^x+x}} \, dx &=\int \left (\frac{x}{\sqrt{e^x+x}}+\frac{e^x x}{\sqrt{e^x+x}}\right ) \, 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}-2 \int \sqrt{e^x+x} \, dx+\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}-2 \int \sqrt{e^x+x} \, dx\\ \end{align*}
Mathematica [A] time = 0.121931, size = 0, normalized size = 0. \[ \int \frac{\left (1+e^x\right ) x}{\sqrt{e^x+x}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 1+{{\rm e}^{x}} \right ) x{\frac{1}{\sqrt{{{\rm e}^{x}}+x}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x{\left (e^{x} + 1\right )}}{\sqrt{x + e^{x}}}\,{d 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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (e^{x} + 1\right )}{\sqrt{x + e^{x}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x{\left (e^{x} + 1\right )}}{\sqrt{x + e^{x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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