Integrand size = 16, antiderivative size = 16 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=2 x \sqrt {e^x+x}-2 \text {Int}\left (\sqrt {e^x+x},x\right ) \]
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Not integrable
Time = 0.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \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*}
Not integrable
Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75
\[\int \frac {\left (1+{\mathrm e}^{x}\right ) x}{\sqrt {{\mathrm e}^{x}+x}}d x\]
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Exception generated. \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 0.74 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\int \frac {x \left (e^{x} + 1\right )}{\sqrt {x + e^{x}}}\, dx \]
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Not integrable
Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\int { \frac {x {\left (e^{x} + 1\right )}}{\sqrt {x + e^{x}}} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\int { \frac {x {\left (e^{x} + 1\right )}}{\sqrt {x + e^{x}}} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\int \frac {x\,\left ({\mathrm {e}}^x+1\right )}{\sqrt {x+{\mathrm {e}}^x}} \,d x \]
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