Optimal. Leaf size=17 \[ 3 \left (x+e^x\right )^{2/3} x+3 \log (x) \]
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Rubi [A] time = 0.64, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {6742, 2261, 2273, 2262} \[ 3 \left (x+e^x\right )^{2/3} x+3 \log (x) \]
Antiderivative was successfully verified.
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Rule 2261
Rule 2262
Rule 2273
Rule 6742
Rubi steps
\begin {align*} \int \frac {5 x^2+3 \sqrt [3]{e^x+x}+e^x \left (3 x+2 x^2\right )}{x \sqrt [3]{e^x+x}} \, dx &=\int \left (\frac {3}{x}+\frac {3 e^x}{\sqrt [3]{e^x+x}}+\frac {\left (5+2 e^x\right ) x}{\sqrt [3]{e^x+x}}\right ) \, dx\\ &=3 \log (x)+3 \int \frac {e^x}{\sqrt [3]{e^x+x}} \, dx+\int \frac {\left (5+2 e^x\right ) x}{\sqrt [3]{e^x+x}} \, dx\\ &=\frac {9}{2} \left (e^x+x\right )^{2/3}+3 \log (x)-3 \int \frac {1}{\sqrt [3]{e^x+x}} \, dx+\int \left (\frac {5 x}{\sqrt [3]{e^x+x}}+\frac {2 e^x x}{\sqrt [3]{e^x+x}}\right ) \, dx\\ &=\frac {9}{2} \left (e^x+x\right )^{2/3}+3 \log (x)+2 \int \frac {e^x x}{\sqrt [3]{e^x+x}} \, dx-3 \int \frac {1}{\sqrt [3]{e^x+x}} \, dx+5 \int \frac {x}{\sqrt [3]{e^x+x}} \, dx\\ &=-3 \left (e^x+x\right )^{2/3}+3 x \left (e^x+x\right )^{2/3}+3 \log (x)-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}+3 \log (x)-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 = 0.24, size = 17, normalized size = 1.00 \[ 3 \left (x+e^x\right )^{2/3} x+3 \log (x) \]
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 \[ \int \frac {5 \, x^{2} + {\left (2 \, x^{2} + 3 \, x\right )} e^{x} + 3 \, {\left (x + e^{x}\right )}^{\frac {1}{3}}}{{\left (x + e^{x}\right )}^{\frac {1}{3}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {5 x^{2}+\left (2 x^{2}+3 x \right ) {\mathrm e}^{x}+3 \left (x +{\mathrm e}^{x}\right )^{\frac {1}{3}}}{\left (x +{\mathrm e}^{x}\right )^{\frac {1}{3}} x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 21, normalized size = 1.24 \[ \frac {3 \, {\left (x^{2} + x e^{x}\right )}}{{\left (x + e^{x}\right )}^{\frac {1}{3}}} + 3 \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.67, size = 14, normalized size = 0.82 \[ 3\,\ln \relax (x)+3\,x\,{\left (x+{\mathrm {e}}^x\right )}^{2/3} \]
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 \[ \int \frac {2 x^{2} e^{x} + 5 x^{2} + 3 x e^{x} + 3 \sqrt [3]{x + e^{x}}}{x \sqrt [3]{x + e^{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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