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

Optimal result

Integrand size = 41, antiderivative size = 28 \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=e^{\frac {x}{2+x^2}} \left (2+x^2\right )+\operatorname {ExpIntegralEi}\left (\frac {x}{2+x^2}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=\int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx \]

[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*} \text {integral}& = \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 [A] (verified)

Time = 0.59 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=2 e^{\frac {x}{2+x^2}}+e^{\frac {x}{2+x^2}} x^2+\operatorname {ExpIntegralEi}\left (\frac {x}{2+x^2}\right ) \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (2 x^{4}-x^{3}+3 x^{2}+2 x +2\right ) {\mathrm e}^{\frac {x}{x^{2}+2}}}{x^{3}+2 x}d x\]

[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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx={\left (x^{2} + 2\right )} e^{\left (\frac {x}{x^{2} + 2}\right )} + {\rm Ei}\left (\frac {x}{x^{2} + 2}\right ) \]

[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))

Sympy [F]

\[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=\int \frac {\left (2 x^{4} - x^{3} + 3 x^{2} + 2 x + 2\right ) e^{\frac {x}{x^{2} + 2}}}{x \left (x^{2} + 2\right )}\, dx \]

[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)

Maxima [F]

\[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=\int { \frac {{\left (2 \, x^{4} - x^{3} + 3 \, x^{2} + 2 \, x + 2\right )} e^{\left (\frac {x}{x^{2} + 2}\right )}}{x^{3} + 2 \, x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=\int { \frac {{\left (2 \, x^{4} - x^{3} + 3 \, x^{2} + 2 \, x + 2\right )} e^{\left (\frac {x}{x^{2} + 2}\right )}}{x^{3} + 2 \, x} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2+2 x+3 x^2-x^3+2 x^4\right )}{2 x+x^3} \, dx=\mathrm {ei}\left (\frac {x}{x^2+2}\right )+2\,{\mathrm {e}}^{\frac {x}{x^2+2}}+x^2\,{\mathrm {e}}^{\frac {x}{x^2+2}} \]

[In]

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

[Out]

ei(x/(x^2 + 2)) + 2*exp(x/(x^2 + 2)) + x^2*exp(x/(x^2 + 2))