3.88.20 \(\int \frac {e^{\frac {2 (16-10 x+x^2-4 x^3-x^4)}{x^2}} (-64+20 x-8 x^3-4 x^4)}{x^3} \, dx\)

Optimal. Leaf size=26 \[ e^{\frac {2 (4-x)^2}{x^2}-\frac {4}{x}-2 x (4+x)} \]

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Rubi [A]  time = 0.26, antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 1, number of rules used = 1, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6706} \begin {gather*} e^{\frac {2 \left (-x^4-4 x^3+x^2-10 x+16\right )}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((2*(16 - 10*x + x^2 - 4*x^3 - x^4))/x^2)*(-64 + 20*x - 8*x^3 - 4*x^4))/x^3,x]

[Out]

E^((2*(16 - 10*x + x^2 - 4*x^3 - x^4))/x^2)

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 {2 \left (16-10 x+x^2-4 x^3-x^4\right )}{x^2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 22, normalized size = 0.85 \begin {gather*} e^{2+\frac {32}{x^2}-\frac {20}{x}-8 x-2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*(16 - 10*x + x^2 - 4*x^3 - x^4))/x^2)*(-64 + 20*x - 8*x^3 - 4*x^4))/x^3,x]

[Out]

E^(2 + 32/x^2 - 20/x - 8*x - 2*x^2)

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fricas [A]  time = 0.52, size = 24, normalized size = 0.92 \begin {gather*} e^{\left (-\frac {2 \, {\left (x^{4} + 4 \, x^{3} - x^{2} + 10 \, x - 16\right )}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^4-8*x^3+20*x-64)*exp((-x^4-4*x^3+x^2-10*x+16)/x^2)^2/x^3,x, algorithm="fricas")

[Out]

e^(-2*(x^4 + 4*x^3 - x^2 + 10*x - 16)/x^2)

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giac [A]  time = 0.14, size = 21, normalized size = 0.81 \begin {gather*} e^{\left (-2 \, x^{2} - 8 \, x - \frac {20}{x} + \frac {32}{x^{2}} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^4-8*x^3+20*x-64)*exp((-x^4-4*x^3+x^2-10*x+16)/x^2)^2/x^3,x, algorithm="giac")

[Out]

e^(-2*x^2 - 8*x - 20/x + 32/x^2 + 2)

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maple [A]  time = 0.17, size = 25, normalized size = 0.96

method	result	size
risch	\({\mathrm e}^{-\frac {2 \left (x^{4}+4 x^{3}-x^{2}+10 x -16\right )}{x^{2}}}\)	\(25\)
norman	\({\mathrm e}^{\frac {-2 x^{4}-8 x^{3}+2 x^{2}-20 x +32}{x^{2}}}\)	\(26\)
gosper	\({\mathrm e}^{-\frac {2 \left (x^{4}+4 x^{3}-x^{2}+10 x -16\right )}{x^{2}}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^4-8*x^3+20*x-64)*exp((-x^4-4*x^3+x^2-10*x+16)/x^2)^2/x^3,x,method=_RETURNVERBOSE)

[Out]

exp(-2*(x^4+4*x^3-x^2+10*x-16)/x^2)

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maxima [A]  time = 0.44, size = 21, normalized size = 0.81 \begin {gather*} e^{\left (-2 \, x^{2} - 8 \, x - \frac {20}{x} + \frac {32}{x^{2}} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^4-8*x^3+20*x-64)*exp((-x^4-4*x^3+x^2-10*x+16)/x^2)^2/x^3,x, algorithm="maxima")

[Out]

e^(-2*x^2 - 8*x - 20/x + 32/x^2 + 2)

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mupad [B]  time = 5.39, size = 25, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^{-8\,x}\,{\mathrm {e}}^2\,{\mathrm {e}}^{-2\,x^2}\,{\mathrm {e}}^{-\frac {20}{x}}\,{\mathrm {e}}^{\frac {32}{x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(2*(10*x - x^2 + 4*x^3 + x^4 - 16))/x^2)*(8*x^3 - 20*x + 4*x^4 + 64))/x^3,x)

[Out]

exp(-8*x)*exp(2)*exp(-2*x^2)*exp(-20/x)*exp(32/x^2)

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sympy [A]  time = 0.16, size = 24, normalized size = 0.92 \begin {gather*} e^{\frac {2 \left (- x^{4} - 4 x^{3} + x^{2} - 10 x + 16\right )}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**4-8*x**3+20*x-64)*exp((-x**4-4*x**3+x**2-10*x+16)/x**2)**2/x**3,x)

[Out]

exp(2*(-x**4 - 4*x**3 + x**2 - 10*x + 16)/x**2)

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