Optimal. Leaf size=12 \[ 3 x \left (e^x+x\right )^{2/3} \]
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Rubi [A]
time = 0.23, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6874, 2305,
2293, 2294} \begin {gather*} 3 x \left (x+e^x\right )^{2/3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2293
Rule 2294
Rule 2305
Rule 6874
Rubi steps
\begin {align*} \int \frac {5 x+e^x (3+2 x)}{\sqrt [3]{e^x+x}} \, dx &=\int \left (\frac {5 x}{\sqrt [3]{e^x+x}}+\frac {e^x (3+2 x)}{\sqrt [3]{e^x+x}}\right ) \, dx\\ &=5 \int \frac {x}{\sqrt [3]{e^x+x}} \, dx+\int \frac {e^x (3+2 x)}{\sqrt [3]{e^x+x}} \, dx\\ &=-\frac {15}{2} \left (e^x+x\right )^{2/3}+5 \int \frac {1}{\sqrt [3]{e^x+x}} \, dx+5 \int \left (e^x+x\right )^{2/3} \, dx+\int \left (\frac {3 e^x}{\sqrt [3]{e^x+x}}+\frac {2 e^x x}{\sqrt [3]{e^x+x}}\right ) \, dx\\ &=-\frac {15}{2} \left (e^x+x\right )^{2/3}+2 \int \frac {e^x x}{\sqrt [3]{e^x+x}} \, dx+3 \int \frac {e^x}{\sqrt [3]{e^x+x}} \, dx+5 \int \frac {1}{\sqrt [3]{e^x+x}} \, dx+5 \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 {x}{\sqrt [3]{e^x+x}} \, dx-3 \int \frac {1}{\sqrt [3]{e^x+x}} \, dx-3 \int \left (e^x+x\right )^{2/3} \, dx+5 \int \frac {1}{\sqrt [3]{e^x+x}} \, dx+5 \int \left (e^x+x\right )^{2/3} \, dx\\ &=3 x \left (e^x+x\right )^{2/3}-2 \int \frac {1}{\sqrt [3]{e^x+x}} \, dx-2 \int \left (e^x+x\right )^{2/3} \, dx-3 \int \frac {1}{\sqrt [3]{e^x+x}} \, dx-3 \int \left (e^x+x\right )^{2/3} \, dx+5 \int \frac {1}{\sqrt [3]{e^x+x}} \, dx+5 \int \left (e^x+x\right )^{2/3} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 12, normalized size = 1.00 \begin {gather*} 3 x \left (e^x+x\right )^{2/3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 10, normalized size = 0.83
method | result | size |
risch | \(3 x \left ({\mathrm e}^{x}+x \right )^{\frac {2}{3}}\) | \(10\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 16, normalized size = 1.33 \begin {gather*} \frac {3 \, {\left (x^{2} + x e^{x}\right )}}{{\left (x + e^{x}\right )}^{\frac {1}{3}}} \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x e^{x} + 5 x + 3 e^{x}}{\sqrt [3]{x + e^{x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
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 [B]
time = 3.47, size = 9, normalized size = 0.75 \begin {gather*} 3\,x\,{\left (x+{\mathrm {e}}^x\right )}^{2/3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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