3.92.50 \(\int \frac {e^{\frac {-5-x+2 x^2-x^3}{x}} (5+2 x^2-2 x^3)}{x^2} \, dx\)

Optimal. Leaf size=17 \[ e^{-1-\frac {5}{x}+2 x-x^2} \]

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Rubi [A]  time = 0.19, antiderivative size = 18, normalized size of antiderivative = 1.06, number of steps used = 1, number of rules used = 1, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6706} \begin {gather*} e^{-\frac {x^3-2 x^2+x+5}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-((5 + x - 2*x^2 + x^3)/x))

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{-\frac {5+x-2 x^2+x^3}{x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 17, normalized size = 1.00 \begin {gather*} e^{-1-\frac {5}{x}+2 x-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-1 - 5/x + 2*x - x^2)

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fricas [A]  time = 0.84, size = 17, normalized size = 1.00 \begin {gather*} e^{\left (-\frac {x^{3} - 2 \, x^{2} + x + 5}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+2*x^2+5)*exp((-x^3+2*x^2-x-5)/x)/x^2,x, algorithm="fricas")

[Out]

e^(-(x^3 - 2*x^2 + x + 5)/x)

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giac [A]  time = 0.16, size = 16, normalized size = 0.94 \begin {gather*} e^{\left (-x^{2} + 2 \, x - \frac {5}{x} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+2*x^2+5)*exp((-x^3+2*x^2-x-5)/x)/x^2,x, algorithm="giac")

[Out]

e^(-x^2 + 2*x - 5/x - 1)

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maple [A]  time = 0.15, size = 18, normalized size = 1.06

method	result	size
gosper	\({\mathrm e}^{-\frac {x^{3}-2 x^{2}+x +5}{x}}\)	\(18\)
risch	\({\mathrm e}^{-\frac {x^{3}-2 x^{2}+x +5}{x}}\)	\(18\)
norman	\({\mathrm e}^{\frac {-x^{3}+2 x^{2}-x -5}{x}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^3+2*x^2+5)*exp((-x^3+2*x^2-x-5)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.42, size = 16, normalized size = 0.94 \begin {gather*} e^{\left (-x^{2} + 2 \, x - \frac {5}{x} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+2*x^2+5)*exp((-x^3+2*x^2-x-5)/x)/x^2,x, algorithm="maxima")

[Out]

e^(-x^2 + 2*x - 5/x - 1)

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mupad [B]  time = 7.71, size = 19, normalized size = 1.12 \begin {gather*} {\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^{-\frac {5}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(x - 2*x^2 + x^3 + 5)/x)*(2*x^2 - 2*x^3 + 5))/x^2,x)

[Out]

exp(2*x)*exp(-1)*exp(-x^2)*exp(-5/x)

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sympy [A]  time = 0.13, size = 14, normalized size = 0.82 \begin {gather*} e^{\frac {- x^{3} + 2 x^{2} - x - 5}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**3+2*x**2+5)*exp((-x**3+2*x**2-x-5)/x)/x**2,x)

[Out]

exp((-x**3 + 2*x**2 - x - 5)/x)

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