Optimal. Leaf size=28 \[ \text{ExpIntegralEi}\left (\frac{x}{x^2+2}\right )+e^{\frac{x}{x^2+2}} \left (x^2+2\right ) \]
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Rubi [F] time = 0.487433, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{e^{\frac{x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{e^{\frac{x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx &=\int \frac{e^{\frac{x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{x \left (2+x^2\right )} \, dx\\ &=\int \left (-e^{\frac{x}{2+x^2}}+\frac{e^{\frac{x}{2+x^2}}}{x}+2 e^{\frac{x}{2+x^2}} x-\frac{2 e^{\frac{x}{2+x^2}} (-2+x)}{2+x^2}\right ) \, dx\\ &=2 \int e^{\frac{x}{2+x^2}} x \, dx-2 \int \frac{e^{\frac{x}{2+x^2}} (-2+x)}{2+x^2} \, dx-\int e^{\frac{x}{2+x^2}} \, dx+\int \frac{e^{\frac{x}{2+x^2}}}{x} \, dx\\ &=2 \int e^{\frac{x}{2+x^2}} x \, dx-2 \int \left (\frac{\left (-2-2 i \sqrt{2}\right ) e^{\frac{x}{2+x^2}}}{4 \left (i \sqrt{2}-x\right )}+\frac{\left (2-2 i \sqrt{2}\right ) e^{\frac{x}{2+x^2}}}{4 \left (i \sqrt{2}+x\right )}\right ) \, dx-\int e^{\frac{x}{2+x^2}} \, dx+\int \frac{e^{\frac{x}{2+x^2}}}{x} \, dx\\ &=2 \int e^{\frac{x}{2+x^2}} x \, dx-\left (-1-i \sqrt{2}\right ) \int \frac{e^{\frac{x}{2+x^2}}}{i \sqrt{2}-x} \, dx-\left (1-i \sqrt{2}\right ) \int \frac{e^{\frac{x}{2+x^2}}}{i \sqrt{2}+x} \, dx-\int e^{\frac{x}{2+x^2}} \, dx+\int \frac{e^{\frac{x}{2+x^2}}}{x} \, dx\\ \end{align*}
Mathematica [F] time = 0.519859, size = 0, normalized size = 0. \[ \int \frac{e^{\frac{x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{\frac{2\,{x}^{4}-{x}^{3}+3\,{x}^{2}+2\,x+2}{{x}^{3}+2\,x}{{\rm e}^{{\frac{x}{{x}^{2}+2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, x^{4} - x^{3} + 3 \, x^{2} + 2 \, x + 2\right )} e^{\left (\frac{x}{x^{2} + 2}\right )}}{x^{3} + 2 \, x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51594, size = 61, normalized size = 2.18 \begin{align*}{\left (x^{2} + 2\right )} e^{\left (\frac{x}{x^{2} + 2}\right )} +{\rm Ei}\left (\frac{x}{x^{2} + 2}\right ) \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{\left (2 x^{4} - x^{3} + 3 x^{2} + 2 x + 2\right ) e^{\frac{x}{x^{2} + 2}}}{x \left (x^{2} + 2\right )}\, 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{{\left (2 \, x^{4} - x^{3} + 3 \, x^{2} + 2 \, x + 2\right )} e^{\left (\frac{x}{x^{2} + 2}\right )}}{x^{3} + 2 \, x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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