\(\int (6+5 e^{e^x+x}+2 x) \, dx\) [6865]

Optimal result

Integrand size = 14, antiderivative size = 20 \[ \int \left (6+5 e^{e^x+x}+2 x\right ) \, dx=5 \left (-2+e^{e^x}+\frac {1}{5} \left (4+(3+x)^2\right )\right ) \]

[Out]

(3+x)^2-6+5*exp(exp(x))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2320, 2225} \[ \int \left (6+5 e^{e^x+x}+2 x\right ) \, dx=x^2+6 x+5 e^{e^x} \]

[In]

Int[6 + 5*E^(E^x + x) + 2*x,x]

[Out]

5*E^E^x + 6*x + x^2

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \left (6+5 e^{e^x+x}+2 x\right ) \, dx=5 e^{e^x}+6 x+x^2 \]

[In]

Integrate[6 + 5*E^(E^x + x) + 2*x,x]

[Out]

5*E^E^x + 6*x + x^2

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.65

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

[In]

int(5*exp(x)*exp(exp(x))+2*x+6,x,method=_RETURNVERBOSE)

[Out]

x^2+6*x+5*exp(exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \left (6+5 e^{e^x+x}+2 x\right ) \, dx={\left ({\left (x^{2} + 6 \, x\right )} e^{x} + 5 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \]

[In]

integrate(5*exp(x)*exp(exp(x))+2*x+6,x, algorithm="fricas")

[Out]

((x^2 + 6*x)*e^x + 5*e^(x + e^x))*e^(-x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.60 \[ \int \left (6+5 e^{e^x+x}+2 x\right ) \, dx=x^{2} + 6 x + 5 e^{e^{x}} \]

[In]

integrate(5*exp(x)*exp(exp(x))+2*x+6,x)

[Out]

x**2 + 6*x + 5*exp(exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.60 \[ \int \left (6+5 e^{e^x+x}+2 x\right ) \, dx=x^{2} + 6 \, x + 5 \, e^{\left (e^{x}\right )} \]

[In]

integrate(5*exp(x)*exp(exp(x))+2*x+6,x, algorithm="maxima")

[Out]

x^2 + 6*x + 5*e^(e^x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.60 \[ \int \left (6+5 e^{e^x+x}+2 x\right ) \, dx=x^{2} + 6 \, x + 5 \, e^{\left (e^{x}\right )} \]

[In]

integrate(5*exp(x)*exp(exp(x))+2*x+6,x, algorithm="giac")

[Out]

x^2 + 6*x + 5*e^(e^x)

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.60 \[ \int \left (6+5 e^{e^x+x}+2 x\right ) \, dx=6\,x+5\,{\mathrm {e}}^{{\mathrm {e}}^x}+x^2 \]

[In]

int(2*x + 5*exp(exp(x))*exp(x) + 6,x)

[Out]

6*x + 5*exp(exp(x)) + x^2