3.98.89 \(\int \frac {1}{30} (-81+30 e^{e^x+x}-50 x) \, dx\)

Optimal. Leaf size=28 \[ -1+e^{e^x}+\frac {1}{5} \left (-e^4-x\right )-\frac {5}{6} x (3+x) \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 0.64, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 2282, 2194} \begin {gather*} -\frac {5 x^2}{6}-\frac {27 x}{10}+e^{e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-81 + 30*E^(E^x + x) - 50*x)/30,x]

[Out]

E^E^x - (27*x)/10 - (5*x^2)/6

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{30} \int \left (-81+30 e^{e^x+x}-50 x\right ) \, dx\\ &=-\frac {27 x}{10}-\frac {5 x^2}{6}+\int e^{e^x+x} \, dx\\ &=-\frac {27 x}{10}-\frac {5 x^2}{6}+\operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=e^{e^x}-\frac {27 x}{10}-\frac {5 x^2}{6}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.64 \begin {gather*} e^{e^x}-\frac {27 x}{10}-\frac {5 x^2}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-81 + 30*E^(E^x + x) - 50*x)/30,x]

[Out]

E^E^x - (27*x)/10 - (5*x^2)/6

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fricas [A]  time = 0.82, size = 26, normalized size = 0.93 \begin {gather*} -\frac {1}{30} \, {\left ({\left (25 \, x^{2} + 81 \, x\right )} e^{x} - 30 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*exp(exp(x))-5/3*x-27/10,x, algorithm="fricas")

[Out]

-1/30*((25*x^2 + 81*x)*e^x - 30*e^(x + e^x))*e^(-x)

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giac [A]  time = 0.16, size = 12, normalized size = 0.43 \begin {gather*} -\frac {5}{6} \, x^{2} - \frac {27}{10} \, x + e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*exp(exp(x))-5/3*x-27/10,x, algorithm="giac")

[Out]

-5/6*x^2 - 27/10*x + e^(e^x)

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maple [A]  time = 0.02, size = 13, normalized size = 0.46

method	result	size
default	\(-\frac {27 x}{10}+{\mathrm e}^{{\mathrm e}^{x}}-\frac {5 x^{2}}{6}\)	\(13\)
norman	\(-\frac {27 x}{10}+{\mathrm e}^{{\mathrm e}^{x}}-\frac {5 x^{2}}{6}\)	\(13\)
risch	\(-\frac {27 x}{10}+{\mathrm e}^{{\mathrm e}^{x}}-\frac {5 x^{2}}{6}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*exp(exp(x))-5/3*x-27/10,x,method=_RETURNVERBOSE)

[Out]

-27/10*x+exp(exp(x))-5/6*x^2

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maxima [A]  time = 0.37, size = 12, normalized size = 0.43 \begin {gather*} -\frac {5}{6} \, x^{2} - \frac {27}{10} \, x + e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*exp(exp(x))-5/3*x-27/10,x, algorithm="maxima")

[Out]

-5/6*x^2 - 27/10*x + e^(e^x)

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mupad [B]  time = 5.63, size = 12, normalized size = 0.43 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^x}-\frac {27\,x}{10}-\frac {5\,x^2}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(x))*exp(x) - (5*x)/3 - 27/10,x)

[Out]

exp(exp(x)) - (27*x)/10 - (5*x^2)/6

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sympy [A]  time = 0.10, size = 15, normalized size = 0.54 \begin {gather*} - \frac {5 x^{2}}{6} - \frac {27 x}{10} + e^{e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)*exp(exp(x))-5/3*x-27/10,x)

[Out]

-5*x**2/6 - 27*x/10 + exp(exp(x))

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