Optimal. Leaf size=52 \[ -\text {Int}\left (\frac {1}{\sqrt {x+e^x}},x\right )-3 \text {Int}\left (\sqrt {x+e^x},x\right )+2 \sqrt {x+e^x} x+2 \sqrt {x+e^x} \]
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Rubi [A] time = 0.08, 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 = {} \[ \int \frac {e^x x}{\sqrt {e^x+x}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {e^x x}{\sqrt {e^x+x}} \, dx &=2 x \sqrt {e^x+x}-2 \int \sqrt {e^x+x} \, dx-\int \frac {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 \sqrt {e^x+x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {e^x x}{\sqrt {e^x+x}} \, dx \]
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 \[ \int \frac {x e^{x}}{\sqrt {x + e^{x}}}\,{d x} \]
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 \[ \int \frac {x \,{\mathrm e}^{x}}{\sqrt {x +{\mathrm e}^{x}}}\, dx \]
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 \[ \int \frac {x e^{x}}{\sqrt {x + e^{x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,{\mathrm {e}}^x}{\sqrt {x+{\mathrm {e}}^x}} \,d x \]
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 \[ \int \frac {x e^{x}}{\sqrt {x + e^{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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