Optimal. Leaf size=28 \[ \frac{x^3}{3}+2 e^x x-2 e^x+\frac{e^{2 x}}{2} \]
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Rubi [A] time = 0.0196767, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6742, 2194, 2176} \[ \frac{x^3}{3}+2 e^x x-2 e^x+\frac{e^{2 x}}{2} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2194
Rule 2176
Rubi steps
\begin{align*} \int \left (e^x+x\right )^2 \, dx &=\int \left (e^{2 x}+2 e^x x+x^2\right ) \, dx\\ &=\frac{x^3}{3}+2 \int e^x x \, dx+\int e^{2 x} \, dx\\ &=\frac{e^{2 x}}{2}+2 e^x x+\frac{x^3}{3}-2 \int e^x \, dx\\ &=-2 e^x+\frac{e^{2 x}}{2}+2 e^x x+\frac{x^3}{3}\\ \end{align*}
Mathematica [A] time = 0.0140445, size = 26, normalized size = 0.93 \[ \frac{x^3}{3}+\frac{e^{2 x}}{2}+e^x (2 x-2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{3}}+{\frac{ \left ({{\rm e}^{x}} \right ) ^{2}}{2}}+2\,{{\rm e}^{x}}x-2\,{{\rm e}^{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.928451, size = 26, normalized size = 0.93 \begin{align*} \frac{1}{3} \, x^{3} + 2 \,{\left (x - 1\right )} e^{x} + \frac{1}{2} \, e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80161, size = 53, normalized size = 1.89 \begin{align*} \frac{1}{3} \, x^{3} + 2 \,{\left (x - 1\right )} e^{x} + \frac{1}{2} \, e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.08726, size = 20, normalized size = 0.71 \begin{align*} \frac{x^{3}}{3} + \frac{\left (4 x - 4\right ) e^{x}}{2} + \frac{e^{2 x}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09778, size = 26, normalized size = 0.93 \begin{align*} \frac{1}{3} \, x^{3} + 2 \,{\left (x - 1\right )} e^{x} + \frac{1}{2} \, e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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