3.66.25 \(\int \frac {1}{135} e^{\frac {1}{27} (-54+27 e^{e^{\frac {5+x}{5}} x}+324 x+x^4)} (1620+20 x^3+e^{e^{\frac {5+x}{5}} x+\frac {5+x}{5}} (135+27 x)) \, dx\)

Optimal. Leaf size=29 \[ e^{-2+e^{e^{1+\frac {x}{5}} x}+3 x \left (4+\frac {x^3}{81}\right )} \]

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Rubi [A]  time = 0.38, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {12, 6706} \begin {gather*} e^{\frac {1}{27} \left (x^4+324 x+27 e^{e^{\frac {x+5}{5}} x}-54\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-54 + 27*E^(E^((5 + x)/5)*x) + 324*x + x^4)/27)*(1620 + 20*x^3 + E^(E^((5 + x)/5)*x + (5 + x)/5)*(135
+ 27*x)))/135,x]

[Out]

E^((-54 + 27*E^(E^((5 + x)/5)*x) + 324*x + x^4)/27)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {1}{135} \int e^{\frac {1}{27} \left (-54+27 e^{e^{\frac {5+x}{5}} x}+324 x+x^4\right )} \left (1620+20 x^3+e^{e^{\frac {5+x}{5}} x+\frac {5+x}{5}} (135+27 x)\right ) \, dx\\ &=e^{\frac {1}{27} \left (-54+27 e^{e^{\frac {5+x}{5}} x}+324 x+x^4\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.78, size = 27, normalized size = 0.93 \begin {gather*} e^{-2+e^{e^{1+\frac {x}{5}} x}+12 x+\frac {x^4}{27}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-54 + 27*E^(E^((5 + x)/5)*x) + 324*x + x^4)/27)*(1620 + 20*x^3 + E^(E^((5 + x)/5)*x + (5 + x)/5
)*(135 + 27*x)))/135,x]

[Out]

E^(-2 + E^(E^(1 + x/5)*x) + 12*x + x^4/27)

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fricas [B]  time = 1.16, size = 41, normalized size = 1.41 \begin {gather*} e^{\left (\frac {1}{27} \, {\left ({\left (x^{4} + 324 \, x - 54\right )} e^{\left (\frac {1}{5} \, x + 1\right )} + 27 \, e^{\left (x e^{\left (\frac {1}{5} \, x + 1\right )} + \frac {1}{5} \, x + 1\right )}\right )} e^{\left (-\frac {1}{5} \, x - 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/135*((27*x+135)*exp(1+1/5*x)*exp(x*exp(1+1/5*x))+20*x^3+1620)*exp(exp(x*exp(1+1/5*x))+1/27*x^4+12*
x-2),x, algorithm="fricas")

[Out]

e^(1/27*((x^4 + 324*x - 54)*e^(1/5*x + 1) + 27*e^(x*e^(1/5*x + 1) + 1/5*x + 1))*e^(-1/5*x - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{135} \, {\left (20 \, x^{3} + 27 \, {\left (x + 5\right )} e^{\left (x e^{\left (\frac {1}{5} \, x + 1\right )} + \frac {1}{5} \, x + 1\right )} + 1620\right )} e^{\left (\frac {1}{27} \, x^{4} + 12 \, x + e^{\left (x e^{\left (\frac {1}{5} \, x + 1\right )}\right )} - 2\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/135*((27*x+135)*exp(1+1/5*x)*exp(x*exp(1+1/5*x))+20*x^3+1620)*exp(exp(x*exp(1+1/5*x))+1/27*x^4+12*
x-2),x, algorithm="giac")

[Out]

integrate(1/135*(20*x^3 + 27*(x + 5)*e^(x*e^(1/5*x + 1) + 1/5*x + 1) + 1620)*e^(1/27*x^4 + 12*x + e^(x*e^(1/5*
x + 1)) - 2), x)

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maple [A]  time = 0.08, size = 21, normalized size = 0.72

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{x \,{\mathrm e}^{1+\frac {x}{5}}}+\frac {x^{4}}{27}+12 x -2}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/135*((27*x+135)*exp(1+1/5*x)*exp(x*exp(1+1/5*x))+20*x^3+1620)*exp(exp(x*exp(1+1/5*x))+1/27*x^4+12*x-2),x
,method=_RETURNVERBOSE)

[Out]

exp(exp(x*exp(1+1/5*x))+1/27*x^4+12*x-2)

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maxima [A]  time = 0.46, size = 20, normalized size = 0.69 \begin {gather*} e^{\left (\frac {1}{27} \, x^{4} + 12 \, x + e^{\left (x e^{\left (\frac {1}{5} \, x + 1\right )}\right )} - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/135*((27*x+135)*exp(1+1/5*x)*exp(x*exp(1+1/5*x))+20*x^3+1620)*exp(exp(x*exp(1+1/5*x))+1/27*x^4+12*
x-2),x, algorithm="maxima")

[Out]

e^(1/27*x^4 + 12*x + e^(x*e^(1/5*x + 1)) - 2)

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mupad [B]  time = 0.21, size = 23, normalized size = 0.79 \begin {gather*} {\mathrm {e}}^{12\,x}\,{\mathrm {e}}^{{\mathrm {e}}^{x\,{\mathrm {e}}^{x/5}\,\mathrm {e}}}\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^{\frac {x^4}{27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(12*x + exp(x*exp(x/5 + 1)) + x^4/27 - 2)*(20*x^3 + exp(x/5 + 1)*exp(x*exp(x/5 + 1))*(27*x + 135) + 16
20))/135,x)

[Out]

exp(12*x)*exp(exp(x*exp(x/5)*exp(1)))*exp(-2)*exp(x^4/27)

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sympy [A]  time = 0.42, size = 20, normalized size = 0.69 \begin {gather*} e^{\frac {x^{4}}{27} + 12 x + e^{x e^{\frac {x}{5} + 1}} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/135*((27*x+135)*exp(1+1/5*x)*exp(x*exp(1+1/5*x))+20*x**3+1620)*exp(exp(x*exp(1+1/5*x))+1/27*x**4+1
2*x-2),x)

[Out]

exp(x**4/27 + 12*x + exp(x*exp(x/5 + 1)) - 2)

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