Optimal. Leaf size=28 \[ \text {Ei}\left (\frac {x}{x^2+2}\right )+e^{\frac {x}{x^2+2}} \left (x^2+2\right ) \]
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Rubi [F] time = 0.49, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}
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Mathematica [A] time = 0.39, size = 39, normalized size = 1.39 \[ \text {Ei}\left (\frac {x}{x^2+2}\right )+e^{\frac {x}{x^2+2}} x^2+2 e^{\frac {x}{x^2+2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 27, normalized size = 0.96 \[ {\left (x^{2} + 2\right )} e^{\left (\frac {x}{x^{2} + 2}\right )} + {\rm Ei}\left (\frac {x}{x^{2} + 2}\right ) \]
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 \[ \int \frac {{\left (2 \, x^{4} - x^{3} + 3 \, x^{2} + 2 \, x + 2\right )} e^{\left (\frac {x}{x^{2} + 2}\right )}}{x^{3} + 2 \, x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (2 x^{4}-x^{3}+3 x^{2}+2 x +2\right ) {\mathrm e}^{\frac {x}{x^{2}+2}}}{x^{3}+2 x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, x^{4} - x^{3} + 3 \, x^{2} + 2 \, x + 2\right )} e^{\left (\frac {x}{x^{2} + 2}\right )}}{x^{3} + 2 \, x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 37, normalized size = 1.32 \[ \mathrm {ei}\left (\frac {x}{x^2+2}\right )+2\,{\mathrm {e}}^{\frac {x}{x^2+2}}+x^2\,{\mathrm {e}}^{\frac {x}{x^2+2}} \]
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 \[ \int \frac {\left (2 x^{4} - x^{3} + 3 x^{2} + 2 x + 2\right ) e^{\frac {x}{x^{2} + 2}}}{x \left (x^{2} + 2\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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