Optimal. Leaf size=17 \[ 3 \left (x+e^x\right )^{2/3} x+3 \log (x) \]
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Rubi [A] time = 0.635028, antiderivative size = 17, normalized size of antiderivative = 1., 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 6742
Rule 2261
Rule 2273
Rule 2262
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*}
Mathematica [A] time = 0.220314, size = 17, normalized size = 1. \[ 3 \left (x+e^x\right )^{2/3} x+3 \log (x) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( 5\,{x}^{2}+3\,\sqrt [3]{{{\rm e}^{x}}+x}+{{\rm e}^{x}} \left ( 2\,{x}^{2}+3\,x \right ) \right ){\frac{1}{\sqrt [3]{{{\rm e}^{x}}+x}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09234, size = 28, normalized size = 1.65 \begin{align*} \frac{3 \,{\left (x^{2} + x e^{x}\right )}}{{\left (x + e^{x}\right )}^{\frac{1}{3}}} + 3 \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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