Optimal. Leaf size=27 \[ 2 x \sqrt {x+e^x}-2 \text {Int}\left (\sqrt {x+e^x},x\right ) \]
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Rubi [A] time = 0.18, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}
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Mathematica [A] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x {\left (e^{x} + 1\right )}}{\sqrt {x + e^{x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left ({\mathrm e}^{x}+1\right ) x}{\sqrt {x +{\mathrm e}^{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x {\left (e^{x} + 1\right )}}{\sqrt {x + e^{x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x\,\left ({\mathrm {e}}^x+1\right )}{\sqrt {x+{\mathrm {e}}^x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (e^{x} + 1\right )}{\sqrt {x + e^{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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