Optimal. Leaf size=28 \[ -2 e^x+\frac {e^{2 x}}{2}+2 e^x x+\frac {x^3}{3} \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 = {6874, 2225,
2207} \begin {gather*} \frac {x^3}{3}+2 e^x x-2 e^x+\frac {e^{2 x}}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 6874
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*}
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Mathematica [A]
time = 0.03, size = 26, normalized size = 0.93 \begin {gather*} \frac {e^{2 x}}{2}+\frac {x^3}{3}+e^x (-2+2 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 22, normalized size = 0.79
method | result | size |
risch | \(\frac {x^{3}}{3}+\left (2 x -2\right ) {\mathrm e}^{x}+\frac {{\mathrm e}^{2 x}}{2}\) | \(21\) |
default | \(-2 \,{\mathrm e}^{x}+\frac {{\mathrm e}^{2 x}}{2}+2 \,{\mathrm e}^{x} x +\frac {x^{3}}{3}\) | \(22\) |
norman | \(-2 \,{\mathrm e}^{x}+\frac {{\mathrm e}^{2 x}}{2}+2 \,{\mathrm e}^{x} x +\frac {x^{3}}{3}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.49, size = 19, normalized size = 0.68 \begin {gather*} \frac {1}{3} \, x^{3} + 2 \, {\left (x - 1\right )} e^{x} + \frac {1}{2} \, e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.09, size = 19, normalized size = 0.68 \begin {gather*} \frac {1}{3} \, x^{3} + 2 \, {\left (x - 1\right )} e^{x} + \frac {1}{2} \, e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 20, normalized size = 0.71 \begin {gather*} \frac {x^{3}}{3} + \frac {\left (4 x - 4\right ) e^{x}}{2} + \frac {e^{2 x}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 19, normalized size = 0.68 \begin {gather*} \frac {1}{3} \, x^{3} + 2 \, {\left (x - 1\right )} e^{x} + \frac {1}{2} \, e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 21, normalized size = 0.75 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}}{2}-2\,{\mathrm {e}}^x+2\,x\,{\mathrm {e}}^x+\frac {x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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