\(\int \frac {e^{\frac {4-x^2}{x}} (24+24 x+6 x^3+e (12+3 x^2))}{5 x^2} \, dx\) [2582]

Optimal result

Integrand size = 39, antiderivative size = 28 \[ \int \frac {e^{\frac {4-x^2}{x}} \left (24+24 x+6 x^3+e \left (12+3 x^2\right )\right )}{5 x^2} \, dx=\frac {3}{5} e^{\frac {4}{x}-x} (-2-e-2 x)-\log (2) \]

[Out]

3/5*(-2*x-exp(1)-2)*exp(4/x-x)-ln(2)

Rubi [F]

\[ \int \frac {e^{\frac {4-x^2}{x}} \left (24+24 x+6 x^3+e \left (12+3 x^2\right )\right )}{5 x^2} \, dx=\int \frac {e^{\frac {4-x^2}{x}} \left (24+24 x+6 x^3+e \left (12+3 x^2\right )\right )}{5 x^2} \, dx \]

[In]

Int[(E^((4 - x^2)/x)*(24 + 24*x + 6*x^3 + E*(12 + 3*x^2)))/(5*x^2),x]

[Out]

(3*Defer[Int][E^(1 + 4/x - x), x])/5 + (12*(2 + E)*Defer[Int][E^(4/x - x)/x^2, x])/5 + (24*Defer[Int][E^(4/x -
x)/x, x])/5 + (6*Defer[Int][E^(4/x - x)*x, x])/5

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \int \frac {e^{\frac {4-x^2}{x}} \left (24+24 x+6 x^3+e \left (12+3 x^2\right )\right )}{x^2} \, dx \\ & = \frac {1}{5} \int \frac {e^{\frac {4}{x}-x} \left (12 (2+e)+24 x+3 e x^2+6 x^3\right )}{x^2} \, dx \\ & = \frac {1}{5} \int \left (3 e^{1+\frac {4}{x}-x}+\frac {12 e^{\frac {4}{x}-x} (2+e)}{x^2}+\frac {24 e^{\frac {4}{x}-x}}{x}+6 e^{\frac {4}{x}-x} x\right ) \, dx \\ & = \frac {3}{5} \int e^{1+\frac {4}{x}-x} \, dx+\frac {6}{5} \int e^{\frac {4}{x}-x} x \, dx+\frac {24}{5} \int \frac {e^{\frac {4}{x}-x}}{x} \, dx+\frac {1}{5} (12 (2+e)) \int \frac {e^{\frac {4}{x}-x}}{x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\frac {4-x^2}{x}} \left (24+24 x+6 x^3+e \left (12+3 x^2\right )\right )}{5 x^2} \, dx=\frac {3}{5} e^{\frac {4}{x}-x} (-2-e-2 x) \]

[In]

Integrate[(E^((4 - x^2)/x)*(24 + 24*x + 6*x^3 + E*(12 + 3*x^2)))/(5*x^2),x]

[Out]

(3*E^(4/x - x)*(-2 - E - 2*x))/5

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75

method	result	size
gosper	\(-\frac {3 \,{\mathrm e}^{-\frac {x^{2}-4}{x}} \left (2 x +{\mathrm e}+2\right )}{5}\)	\(21\)
risch	\(\frac {\left (-3 \,{\mathrm e}-6-6 x \right ) {\mathrm e}^{-\frac {\left (-2+x \right ) \left (2+x \right )}{x}}}{5}\)	\(24\)
norman	\(\frac {\left (-\frac {6}{5}-\frac {3 \,{\mathrm e}}{5}\right ) x \,{\mathrm e}^{\frac {-x^{2}+4}{x}}-\frac {6 x^{2} {\mathrm e}^{\frac {-x^{2}+4}{x}}}{5}}{x}\)	\(43\)
parallelrisch	\(-\frac {3 \,{\mathrm e}^{-\frac {x^{2}-4}{x}} {\mathrm e}}{5}-\frac {6 x \,{\mathrm e}^{-\frac {x^{2}-4}{x}}}{5}-\frac {6 \,{\mathrm e}^{-\frac {x^{2}-4}{x}}}{5}\)	\(44\)

[In]

int(1/5*((3*x^2+12)*exp(1)+6*x^3+24*x+24)*exp((-x^2+4)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

-3/5*exp(-(x^2-4)/x)*(2*x+exp(1)+2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {4-x^2}{x}} \left (24+24 x+6 x^3+e \left (12+3 x^2\right )\right )}{5 x^2} \, dx=-\frac {3}{5} \, {\left (2 \, x + e + 2\right )} e^{\left (-\frac {x^{2} - 4}{x}\right )} \]

[In]

integrate(1/5*((3*x^2+12)*exp(1)+6*x^3+24*x+24)*exp((-x^2+4)/x)/x^2,x, algorithm="fricas")

[Out]

-3/5*(2*x + e + 2)*e^(-(x^2 - 4)/x)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {4-x^2}{x}} \left (24+24 x+6 x^3+e \left (12+3 x^2\right )\right )}{5 x^2} \, dx=\frac {\left (- 6 x - 3 e - 6\right ) e^{\frac {4 - x^{2}}{x}}}{5} \]

[In]

integrate(1/5*((3*x**2+12)*exp(1)+6*x**3+24*x+24)*exp((-x**2+4)/x)/x**2,x)

[Out]

(-6*x - 3*E - 6)*exp((4 - x**2)/x)/5

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {e^{\frac {4-x^2}{x}} \left (24+24 x+6 x^3+e \left (12+3 x^2\right )\right )}{5 x^2} \, dx=-\frac {3}{5} \, {\left (2 \, x + e + 2\right )} e^{\left (-x + \frac {4}{x}\right )} \]

[In]

integrate(1/5*((3*x^2+12)*exp(1)+6*x^3+24*x+24)*exp((-x^2+4)/x)/x^2,x, algorithm="maxima")

[Out]

-3/5*(2*x + e + 2)*e^(-x + 4/x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {e^{\frac {4-x^2}{x}} \left (24+24 x+6 x^3+e \left (12+3 x^2\right )\right )}{5 x^2} \, dx=-\frac {6}{5} \, x e^{\left (-\frac {x^{2} - 4}{x}\right )} - \frac {3}{5} \, e^{\left (-\frac {x^{2} - x - 4}{x}\right )} - \frac {6}{5} \, e^{\left (-\frac {x^{2} - 4}{x}\right )} \]

[In]

integrate(1/5*((3*x^2+12)*exp(1)+6*x^3+24*x+24)*exp((-x^2+4)/x)/x^2,x, algorithm="giac")

[Out]

-6/5*x*e^(-(x^2 - 4)/x) - 3/5*e^(-(x^2 - x - 4)/x) - 6/5*e^(-(x^2 - 4)/x)

Mupad [B] (verification not implemented)

Time = 7.95 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75 \[ \int \frac {e^{\frac {4-x^2}{x}} \left (24+24 x+6 x^3+e \left (12+3 x^2\right )\right )}{5 x^2} \, dx=-{\mathrm {e}}^{\frac {4}{x}-x}\,\left (\frac {6\,x}{5}+\frac {3\,\mathrm {e}}{5}+\frac {6}{5}\right ) \]

[In]

int((exp(-(x^2 - 4)/x)*(24*x + exp(1)*(3*x^2 + 12) + 6*x^3 + 24))/(5*x^2),x)

[Out]

-exp(4/x - x)*((6*x)/5 + (3*exp(1))/5 + 6/5)