Integrand size = 37, antiderivative size = 12 \[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=2 x \sqrt {e^x+x} \]
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Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2305, 2294} \[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=2 x \sqrt {x+e^x} \]
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Rule 2294
Rule 2305
Rubi steps \begin{align*} \text {integral}& = 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*}
Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=2 x \sqrt {e^x+x} \]
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Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83
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Exception generated. \[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=\int \frac {x e^{x} + 3 x + 2 e^{x}}{\sqrt {x + e^{x}}}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=2 \, \sqrt {x + e^{x}} x \]
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\[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=\int { \frac {x e^{x}}{\sqrt {x + e^{x}}} + 2 \, \sqrt {x + e^{x}} + \frac {x}{\sqrt {x + e^{x}}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}+2 \sqrt {e^x+x}\right ) \, dx=2\,x\,\sqrt {x+{\mathrm {e}}^x} \]
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