Optimal. Leaf size=12 \[ 2 x \sqrt {x+e^x} \]
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Rubi [A] time = 0.26, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 2262
Rule 2273
Rule 6742
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*}
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Mathematica [A] time = 0.10, size = 12, normalized size = 1.00 \[ 2 x \sqrt {x+e^x} \]
Antiderivative was successfully verified.
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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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x {\left (e^{x} + 1\right )}}{\sqrt {x + e^{x}}} + 2 \, \sqrt {x + e^{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 10, normalized size = 0.83 \[ 2 \sqrt {x +{\mathrm e}^{x}}\, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 16, normalized size = 1.33 \[ \frac {2 \, {\left (x^{2} + x e^{x}\right )}}{\sqrt {x + e^{x}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 9, normalized size = 0.75 \[ 2\,x\,\sqrt {x+{\mathrm {e}}^x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x e^{x} + 3 x + 2 e^{x}}{\sqrt {x + e^{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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