Optimal. Leaf size=12 \[ 3 x \left (x+e^x\right )^{2/3} \]
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Rubi [A] time = 0.143765, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {2273, 2262} \[ 3 x \left (x+e^x\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 2273
Rule 2262
Rubi steps
\begin{align*} \int \left (\frac{2 x}{\sqrt [3]{e^x+x}}+\frac{2 e^x x}{\sqrt [3]{e^x+x}}+3 \left (e^x+x\right )^{2/3}\right ) \, dx &=2 \int \frac{x}{\sqrt [3]{e^x+x}} \, dx+2 \int \frac{e^x x}{\sqrt [3]{e^x+x}} \, dx+3 \int \left (e^x+x\right )^{2/3} \, dx\\ &=-3 \left (e^x+x\right )^{2/3}+3 x \left (e^x+x\right )^{2/3}+2 \int \frac{1}{\sqrt [3]{e^x+x}} \, dx-2 \int \frac{x}{\sqrt [3]{e^x+x}} \, dx+2 \int \left (e^x+x\right )^{2/3} \, dx\\ &=3 x \left (e^x+x\right )^{2/3}\\ \end{align*}
Mathematica [A] time = 0.0470589, size = 12, normalized size = 1. \[ 3 x \left (x+e^x\right )^{2/3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 10, normalized size = 0.8 \begin{align*} 3\,x \left ({{\rm e}^{x}}+x \right ) ^{2/3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1051, size = 22, normalized size = 1.83 \begin{align*} \frac{3 \,{\left (x^{2} + x e^{x}\right )}}{{\left (x + e^{x}\right )}^{\frac{1}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 x e^{x} + 5 x + 3 e^{x}}{\sqrt [3]{x + e^{x}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x e^{x}}{{\left (x + e^{x}\right )}^{\frac{1}{3}}} + 3 \,{\left (x + e^{x}\right )}^{\frac{2}{3}} + \frac{2 \, x}{{\left (x + e^{x}\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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