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

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

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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} $$\begin{array}{c}{x}^2+6x+5e^{e^x}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =6x+x^2+5\int e^{e^x+x}dx\\ & =6x+x^2+5\mathrm{Subst}(\int e^{x}dx,x,e^x)\\ & =5e^{e^x}+6x+x^2\end{array}\end{array}$$

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

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 $$\begin{array}{c}((x^2+6x)e^x+5e^{(x+e^x)})e^{(-x)}\end{array}$$

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 $$\begin{array}{c}{x}^2+6x+5e^{\left(e^x\right)}\end{array}$$

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	$x^2+6x+5{\mathrm{e}}^{{\mathrm{e}}^x}$	$13$
norman	$x^2+6x+5{\mathrm{e}}^{{\mathrm{e}}^x}$	$13$
risch	$x^2+6x+5{\mathrm{e}}^{{\mathrm{e}}^x}$	$13$

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 $$\begin{array}{c}{x}^2+6x+5e^{\left(e^x\right)}\end{array}$$

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 $$\begin{array}{c}6x+5{\mathrm{e}}^{{\mathrm{e}}^x}+x^2\end{array}$$

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 $$\begin{array}{c}{x}^2+6x+5e^{e^x}\end{array}$$

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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