3.66.79 \(\int \frac {e^{\frac {9+4 x-5 x^2}{x}} (-9-5 x^2)}{x^2} \, dx\)

Optimal. Leaf size=17 \[ e^{-5-3 (-3+x)+\frac {9}{x}-2 x} \]

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Rubi [A]  time = 0.14, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6706} \begin {gather*} e^{\frac {-5 x^2+4 x+9}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((9 + 4*x - 5*x^2)/x)*(-9 - 5*x^2))/x^2,x]

[Out]

E^((9 + 4*x - 5*x^2)/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 {9+4 x-5 x^2}{x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 12, normalized size = 0.71 \begin {gather*} e^{4+\frac {9}{x}-5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((9 + 4*x - 5*x^2)/x)*(-9 - 5*x^2))/x^2,x]

[Out]

E^(4 + 9/x - 5*x)

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fricas [A]  time = 0.56, size = 16, normalized size = 0.94 \begin {gather*} e^{\left (-\frac {5 \, x^{2} - 4 \, x - 9}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x^2-9)*exp((-5*x^2+4*x+9)/x)/x^2,x, algorithm="fricas")

[Out]

e^(-(5*x^2 - 4*x - 9)/x)

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giac [A]  time = 0.22, size = 11, normalized size = 0.65 \begin {gather*} e^{\left (-5 \, x + \frac {9}{x} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x^2-9)*exp((-5*x^2+4*x+9)/x)/x^2,x, algorithm="giac")

[Out]

e^(-5*x + 9/x + 4)

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maple [A]  time = 0.15, size = 15, normalized size = 0.88

method	result	size
risch	\({\mathrm e}^{-\frac {\left (x +1\right ) \left (5 x -9\right )}{x}}\)	\(15\)
norman	\({\mathrm e}^{\frac {-5 x^{2}+4 x +9}{x}}\)	\(16\)
gosper	\({\mathrm e}^{-\frac {5 x^{2}-4 x -9}{x}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-5*x^2-9)*exp((-5*x^2+4*x+9)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(-(x+1)*(5*x-9)/x)

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maxima [A]  time = 0.43, size = 11, normalized size = 0.65 \begin {gather*} e^{\left (-5 \, x + \frac {9}{x} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x^2-9)*exp((-5*x^2+4*x+9)/x)/x^2,x, algorithm="maxima")

[Out]

e^(-5*x + 9/x + 4)

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mupad [B]  time = 4.16, size = 13, normalized size = 0.76 \begin {gather*} {\mathrm {e}}^{-5\,x}\,{\mathrm {e}}^4\,{\mathrm {e}}^{9/x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((4*x - 5*x^2 + 9)/x)*(5*x^2 + 9))/x^2,x)

[Out]

exp(-5*x)*exp(4)*exp(9/x)

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sympy [A]  time = 0.11, size = 12, normalized size = 0.71 \begin {gather*} e^{\frac {- 5 x^{2} + 4 x + 9}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-5*x**2-9)*exp((-5*x**2+4*x+9)/x)/x**2,x)

[Out]

exp((-5*x**2 + 4*x + 9)/x)

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