Optimal. Leaf size=12 \[ 3 x \left (x+e^x\right )^{2/3} \]
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Rubi [A] time = 0.34, 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 = {6742, 2273, 2261, 2262} \[ 3 x \left (x+e^x\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 2261
Rule 2262
Rule 2273
Rule 6742
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 = 0.10, size = 12, normalized size = 1.00 \[ 3 x \left (x+e^x\right )^{2/3} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {{\left (2 \, x + 3\right )} e^{x} + 5 \, x}{{\left (x + e^{x}\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 10, normalized size = 0.83 \[ 3 \left (x +{\mathrm e}^{x}\right )^{\frac {2}{3}} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 16, normalized size = 1.33 \[ \frac {3 \, {\left (x^{2} + x e^{x}\right )}}{{\left (x + e^{x}\right )}^{\frac {1}{3}}} \]
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 \[ 3\,x\,{\left (x+{\mathrm {e}}^x\right )}^{2/3} \]
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 \[ \int \frac {2 x e^{x} + 5 x + 3 e^{x}}{\sqrt [3]{x + e^{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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