\(\int \frac {1}{30} (-81+30 e^{e^x+x}-50 x) \, dx\) [9789]

Optimal result

Integrand size = 18, antiderivative size = 28 \[ \int \frac {1}{30} \left (-81+30 e^{e^x+x}-50 x\right ) \, dx=-1+e^{e^x}+\frac {1}{5} \left (-e^4-x\right )-\frac {5}{6} x (3+x) \]

[Out]

-1/5*x-1/6*x*(5*x+15)-1/5*exp(4)+exp(exp(x))-1

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.64, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 2320, 2225} \[ \int \frac {1}{30} \left (-81+30 e^{e^x+x}-50 x\right ) \, dx=-\frac {5 x^2}{6}-\frac {27 x}{10}+e^{e^x} \]

[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 2225

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 2320

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{align*} \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}+\text {Subst}\left (\int e^x \, dx,x,e^x\right ) \\ & = e^{e^x}-\frac {27 x}{10}-\frac {5 x^2}{6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.64 \[ \int \frac {1}{30} \left (-81+30 e^{e^x+x}-50 x\right ) \, dx=e^{e^x}-\frac {27 x}{10}-\frac {5 x^2}{6} \]

[In]

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

[Out]

E^E^x - (27*x)/10 - (5*x^2)/6

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.46

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {1}{30} \left (-81+30 e^{e^x+x}-50 x\right ) \, dx=-\frac {1}{30} \, {\left ({\left (25 \, x^{2} + 81 \, x\right )} e^{x} - 30 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.54 \[ \int \frac {1}{30} \left (-81+30 e^{e^x+x}-50 x\right ) \, dx=- \frac {5 x^{2}}{6} - \frac {27 x}{10} + e^{e^{x}} \]

[In]

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

[Out]

-5*x**2/6 - 27*x/10 + exp(exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.43 \[ \int \frac {1}{30} \left (-81+30 e^{e^x+x}-50 x\right ) \, dx=-\frac {5}{6} \, x^{2} - \frac {27}{10} \, x + e^{\left (e^{x}\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.43 \[ \int \frac {1}{30} \left (-81+30 e^{e^x+x}-50 x\right ) \, dx=-\frac {5}{6} \, x^{2} - \frac {27}{10} \, x + e^{\left (e^{x}\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 17.33 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.43 \[ \int \frac {1}{30} \left (-81+30 e^{e^x+x}-50 x\right ) \, dx={\mathrm {e}}^{{\mathrm {e}}^x}-\frac {27\,x}{10}-\frac {5\,x^2}{6} \]

[In]

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

[Out]

exp(exp(x)) - (27*x)/10 - (5*x^2)/6