Optimal. Leaf size=27 \[ 2 x \sqrt {e^x+x}-2 \text {Int}\left (\sqrt {e^x+x},x\right ) \]
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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 = {}
\begin {gather*} \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \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 = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\frac {x}{\sqrt {e^x+x}}+\frac {e^x x}{\sqrt {e^x+x}}\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {x}{\sqrt {{\mathrm e}^{x}+x}}+\frac {{\mathrm e}^{x} x}{\sqrt {{\mathrm e}^{x}+x}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {x \left (e^{x} + 1\right )}{\sqrt {x + e^{x}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x}{\sqrt {x+{\mathrm {e}}^x}}+\frac {x\,{\mathrm {e}}^x}{\sqrt {x+{\mathrm {e}}^x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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