3.3 \(\int \frac{e^{\frac{x}{2+x^2}} (2+2 x+3 x^2-x^3+2 x^4)}{2 x+x^3} \, dx\)

Optimal. Leaf size=28 \[ \text{ExpIntegralEi}\left (\frac{x}{x^2+2}\right )+e^{\frac{x}{x^2+2}} \left (x^2+2\right ) \]

[Out]

E^(x/(2 + x^2))*(2 + x^2) + ExpIntegralEi[x/(2 + x^2)]

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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.

[In]

Int[(E^(x/(2 + x^2))*(2 + 2*x + 3*x^2 - x^3 + 2*x^4))/(2*x + x^3),x]

[Out]

-Defer[Int][E^(x/(2 + x^2)), x] + (1 + I*Sqrt[2])*Defer[Int][E^(x/(2 + x^2))/(I*Sqrt[2] - x), x] + Defer[Int][
E^(x/(2 + x^2))/x, x] + 2*Defer[Int][E^(x/(2 + x^2))*x, x] - (1 - I*Sqrt[2])*Defer[Int][E^(x/(2 + x^2))/(I*Sqr
t[2] + x), x]

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.

[In]

Integrate[(E^(x/(2 + x^2))*(2 + 2*x + 3*x^2 - x^3 + 2*x^4))/(2*x + x^3),x]

[Out]

Integrate[(E^(x/(2 + x^2))*(2 + 2*x + 3*x^2 - x^3 + 2*x^4))/(2*x + x^3), x]

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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.

[In]

int((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2))/(x^3+2*x),x)

[Out]

int((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2))/(x^3+2*x),x)

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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.

[In]

integrate((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2))/(x^3+2*x),x, algorithm="maxima")

[Out]

integrate((2*x^4 - x^3 + 3*x^2 + 2*x + 2)*e^(x/(x^2 + 2))/(x^3 + 2*x), x)

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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.

[In]

integrate((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2))/(x^3+2*x),x, algorithm="fricas")

[Out]

(x^2 + 2)*e^(x/(x^2 + 2)) + Ei(x/(x^2 + 2))

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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.

[In]

integrate((2*x**4-x**3+3*x**2+2*x+2)*exp(x/(x**2+2))/(x**3+2*x),x)

[Out]

Integral((2*x**4 - x**3 + 3*x**2 + 2*x + 2)*exp(x/(x**2 + 2))/(x*(x**2 + 2)), x)

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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.

[In]

integrate((2*x^4-x^3+3*x^2+2*x+2)*exp(x/(x^2+2))/(x^3+2*x),x, algorithm="giac")

[Out]

integrate((2*x^4 - x^3 + 3*x^2 + 2*x + 2)*e^(x/(x^2 + 2))/(x^3 + 2*x), x)