3.69.65 \(\int (6+5 e^{e^x+x}+2 x) \, dx\)

Optimal. Leaf size=20 \[ 5 \left (-2+e^{e^x}+\frac {1}{5} \left (4+(3+x)^2\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2282, 2194} \begin {gather*} x^2+6 x+5 e^{e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[6 + 5*E^(E^x + x) + 2*x,x]

[Out]

5*E^E^x + 6*x + x^2

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

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.70 \begin {gather*} 5 e^{e^x}+6 x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[6 + 5*E^(E^x + x) + 2*x,x]

[Out]

5*E^E^x + 6*x + x^2

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fricas [A]  time = 0.97, size = 23, normalized size = 1.15 \begin {gather*} {\left ({\left (x^{2} + 6 \, x\right )} e^{x} + 5 \, 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(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)

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giac [A]  time = 0.25, size = 12, normalized size = 0.60 \begin {gather*} x^{2} + 6 \, x + 5 \, e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + 6*x + 5*e^(e^x)

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maple [A]  time = 0.02, size = 13, normalized size = 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	\(x^{2}+6 x +5 \,{\mathrm e}^{{\mathrm e}^{x}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.35, size = 12, normalized size = 0.60 \begin {gather*} x^{2} + 6 \, x + 5 \, e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 + 6*x + 5*e^(e^x)

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mupad [B]  time = 0.05, size = 12, normalized size = 0.60 \begin {gather*} 6\,x+5\,{\mathrm {e}}^{{\mathrm {e}}^x}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + 5*exp(exp(x))*exp(x) + 6,x)

[Out]

6*x + 5*exp(exp(x)) + x^2

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sympy [A]  time = 0.10, size = 12, normalized size = 0.60 \begin {gather*} x^{2} + 6 x + 5 e^{e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2 + 6*x + 5*exp(exp(x))

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