3.85.43 \(\int \frac {e^{e^x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4}}{x^5}} (e^x x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4} (-5-5000 x-6000 x^2-2400 x^3-320 x^4)}{x^5})}{x} \, dx\)

Optimal. Leaf size=22 \[ e^{e^x+\frac {1024 e^{-5 (5+2 x)^4}}{x^5}} \]

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Rubi [A]  time = 0.92, antiderivative size = 33, normalized size of antiderivative = 1.50, number of steps used = 1, number of rules used = 1, integrand size = 90, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {6706} \begin {gather*} \exp \left (\frac {1024 e^{-80 x^4-800 x^3-3000 x^2-5000 x-3125}}{x^5}+e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(E^x + (1024*E^(-3125 - 5000*x - 3000*x^2 - 800*x^3 - 80*x^4))/x^5)*(E^x*x + (1024*E^(-3125 - 5000*x -
3000*x^2 - 800*x^3 - 80*x^4)*(-5 - 5000*x - 6000*x^2 - 2400*x^3 - 320*x^4))/x^5))/x,x]

[Out]

E^(E^x + (1024*E^(-3125 - 5000*x - 3000*x^2 - 800*x^3 - 80*x^4))/x^5)

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} &=\exp \left (e^x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4}}{x^5}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 22, normalized size = 1.00 \begin {gather*} e^{e^x+\frac {1024 e^{-5 (5+2 x)^4}}{x^5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^x + (1024*E^(-3125 - 5000*x - 3000*x^2 - 800*x^3 - 80*x^4))/x^5)*(E^x*x + (1024*E^(-3125 - 500
0*x - 3000*x^2 - 800*x^3 - 80*x^4)*(-5 - 5000*x - 6000*x^2 - 2400*x^3 - 320*x^4))/x^5))/x,x]

[Out]

E^(E^x + 1024/(E^(5*(5 + 2*x)^4)*x^5))

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fricas [A]  time = 0.62, size = 35, normalized size = 1.59 \begin {gather*} e^{\left (\frac {x^{5} e^{x} + 1024 \, e^{\left (-80 \, x^{4} - 800 \, x^{3} - 3000 \, x^{2} - 5000 \, x - 3125\right )}}{x^{5}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1024*(-320*x^4-2400*x^3-6000*x^2-5000*x-5)/x^5/exp(16*x^4+160*x^3+600*x^2+1000*x+625)^5+exp(x)*x)*e
xp(1024/x^5/exp(16*x^4+160*x^3+600*x^2+1000*x+625)^5+exp(x))/x,x, algorithm="fricas")

[Out]

e^((x^5*e^x + 1024*e^(-80*x^4 - 800*x^3 - 3000*x^2 - 5000*x - 3125))/x^5)

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giac [A]  time = 0.36, size = 30, normalized size = 1.36 \begin {gather*} e^{\left (\frac {1024 \, e^{\left (-80 \, x^{4} - 800 \, x^{3} - 3000 \, x^{2} - 5000 \, x - 3125\right )}}{x^{5}} + e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1024*(-320*x^4-2400*x^3-6000*x^2-5000*x-5)/x^5/exp(16*x^4+160*x^3+600*x^2+1000*x+625)^5+exp(x)*x)*e
xp(1024/x^5/exp(16*x^4+160*x^3+600*x^2+1000*x+625)^5+exp(x))/x,x, algorithm="giac")

[Out]

e^(1024*e^(-80*x^4 - 800*x^3 - 3000*x^2 - 5000*x - 3125)/x^5 + e^x)

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maple [A]  time = 0.08, size = 25, normalized size = 1.14

method	result	size
risch	\({\mathrm e}^{\frac {x^{5} {\mathrm e}^{x}+1024 \,{\mathrm e}^{-5 \left (5+2 x \right )^{4}}}{x^{5}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1024*(-320*x^4-2400*x^3-6000*x^2-5000*x-5)/x^5/exp(16*x^4+160*x^3+600*x^2+1000*x+625)^5+exp(x)*x)*exp(102
4/x^5/exp(16*x^4+160*x^3+600*x^2+1000*x+625)^5+exp(x))/x,x,method=_RETURNVERBOSE)

[Out]

exp((x^5*exp(x)+1024*exp(-5*(5+2*x)^4))/x^5)

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maxima [A]  time = 0.61, size = 30, normalized size = 1.36 \begin {gather*} e^{\left (\frac {1024 \, e^{\left (-80 \, x^{4} - 800 \, x^{3} - 3000 \, x^{2} - 5000 \, x - 3125\right )}}{x^{5}} + e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1024*(-320*x^4-2400*x^3-6000*x^2-5000*x-5)/x^5/exp(16*x^4+160*x^3+600*x^2+1000*x+625)^5+exp(x)*x)*e
xp(1024/x^5/exp(16*x^4+160*x^3+600*x^2+1000*x+625)^5+exp(x))/x,x, algorithm="maxima")

[Out]

e^(1024*e^(-80*x^4 - 800*x^3 - 3000*x^2 - 5000*x - 3125)/x^5 + e^x)

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mupad [B]  time = 5.25, size = 34, normalized size = 1.55 \begin {gather*} {\mathrm {e}}^{\frac {1024\,{\mathrm {e}}^{-5000\,x}\,{\mathrm {e}}^{-3125}\,{\mathrm {e}}^{-80\,x^4}\,{\mathrm {e}}^{-800\,x^3}\,{\mathrm {e}}^{-3000\,x^2}}{x^5}}\,{\mathrm {e}}^{{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x) + (1024*exp(- 5000*x - 3000*x^2 - 800*x^3 - 80*x^4 - 3125))/x^5)*(x*exp(x) - (exp(- 5000*x - 3
000*x^2 - 800*x^3 - 80*x^4 - 3125)*(5120000*x + 6144000*x^2 + 2457600*x^3 + 327680*x^4 + 5120))/x^5))/x,x)

[Out]

exp((1024*exp(-5000*x)*exp(-3125)*exp(-80*x^4)*exp(-800*x^3)*exp(-3000*x^2))/x^5)*exp(exp(x))

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sympy [A]  time = 0.60, size = 32, normalized size = 1.45 \begin {gather*} e^{e^{x} + \frac {1024 e^{- 80 x^{4} - 800 x^{3} - 3000 x^{2} - 5000 x - 3125}}{x^{5}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1024*(-320*x**4-2400*x**3-6000*x**2-5000*x-5)/x**5/exp(16*x**4+160*x**3+600*x**2+1000*x+625)**5+exp
(x)*x)*exp(1024/x**5/exp(16*x**4+160*x**3+600*x**2+1000*x+625)**5+exp(x))/x,x)

[Out]

exp(exp(x) + 1024*exp(-80*x**4 - 800*x**3 - 3000*x**2 - 5000*x - 3125)/x**5)

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