Integrand size = 43, antiderivative size = 17 \[ \int \frac {5 x^2+3 \sqrt [3]{e^x+x}+e^x \left (3 x+2 x^2\right )}{x \sqrt [3]{e^x+x}} \, dx=3 x \left (e^x+x\right )^{2/3}+3 \log (x) \]
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Time = 0.44 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {6874, 2293, 2305, 2294} \[ \int \frac {5 x^2+3 \sqrt [3]{e^x+x}+e^x \left (3 x+2 x^2\right )}{x \sqrt [3]{e^x+x}} \, dx=3 \left (x+e^x\right )^{2/3} x+3 \log (x) \]
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Rule 2293
Rule 2294
Rule 2305
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.54 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {5 x^2+3 \sqrt [3]{e^x+x}+e^x \left (3 x+2 x^2\right )}{x \sqrt [3]{e^x+x}} \, dx=3 x \left (e^x+x\right )^{2/3}+3 \log (x) \]
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\[\int \frac {5 x^{2}+3 \left ({\mathrm e}^{x}+x \right )^{\frac {1}{3}}+{\mathrm e}^{x} \left (2 x^{2}+3 x \right )}{x \left ({\mathrm e}^{x}+x \right )^{\frac {1}{3}}}d x\]
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Exception generated. \[ \int \frac {5 x^2+3 \sqrt [3]{e^x+x}+e^x \left (3 x+2 x^2\right )}{x \sqrt [3]{e^x+x}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {5 x^2+3 \sqrt [3]{e^x+x}+e^x \left (3 x+2 x^2\right )}{x \sqrt [3]{e^x+x}} \, dx=\int \frac {2 x^{2} e^{x} + 5 x^{2} + 3 x e^{x} + 3 \sqrt [3]{x + e^{x}}}{x \sqrt [3]{x + e^{x}}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {5 x^2+3 \sqrt [3]{e^x+x}+e^x \left (3 x+2 x^2\right )}{x \sqrt [3]{e^x+x}} \, dx=\frac {3 \, {\left (x^{2} + x e^{x}\right )}}{{\left (x + e^{x}\right )}^{\frac {1}{3}}} + 3 \, \log \left (x\right ) \]
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\[ \int \frac {5 x^2+3 \sqrt [3]{e^x+x}+e^x \left (3 x+2 x^2\right )}{x \sqrt [3]{e^x+x}} \, dx=\int { \frac {5 \, x^{2} + {\left (2 \, x^{2} + 3 \, x\right )} e^{x} + 3 \, {\left (x + e^{x}\right )}^{\frac {1}{3}}}{{\left (x + e^{x}\right )}^{\frac {1}{3}} x} \,d x } \]
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Time = 0.39 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {5 x^2+3 \sqrt [3]{e^x+x}+e^x \left (3 x+2 x^2\right )}{x \sqrt [3]{e^x+x}} \, dx=3\,\ln \left (x\right )+3\,x\,{\left (x+{\mathrm {e}}^x\right )}^{2/3} \]
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