3.100.16 \(\int (1+e^{8/3}+e^{\frac {8}{3}+x}) \, dx\)

Optimal. Leaf size=17 \[ x+e^{8/3} \left (-5+e+e^x+x+\log (4)\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2194} \begin {gather*} \left (1+e^{8/3}\right ) x+e^{x+\frac {8}{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + E^(8/3) + E^(8/3 + x),x]

[Out]

E^(8/3 + x) + (1 + E^(8/3))*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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (1+e^{8/3}\right ) x+\int e^{\frac {8}{3}+x} \, dx\\ &=e^{\frac {8}{3}+x}+\left (1+e^{8/3}\right ) x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.94 \begin {gather*} e^{\frac {8}{3}+x}+x+e^{8/3} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + E^(8/3) + E^(8/3 + x),x]

[Out]

E^(8/3 + x) + x + E^(8/3)*x

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fricas [A]  time = 0.79, size = 10, normalized size = 0.59 \begin {gather*} x e^{\frac {8}{3}} + x + e^{\left (x + \frac {8}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(4/3)^2*exp(x)+exp(4/3)^2+1,x, algorithm="fricas")

[Out]

x*e^(8/3) + x + e^(x + 8/3)

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giac [A]  time = 0.12, size = 10, normalized size = 0.59 \begin {gather*} x e^{\frac {8}{3}} + x + e^{\left (x + \frac {8}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(4/3)^2*exp(x)+exp(4/3)^2+1,x, algorithm="giac")

[Out]

x*e^(8/3) + x + e^(x + 8/3)

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maple [A]  time = 0.03, size = 11, normalized size = 0.65

method	result	size
risch	\(x +{\mathrm e}^{\frac {8}{3}} x +{\mathrm e}^{x +\frac {8}{3}}\)	\(11\)
derivativedivides	\({\mathrm e}^{\frac {8}{3}} {\mathrm e}^{x}+\left ({\mathrm e}^{\frac {8}{3}}+1\right ) \ln \left ({\mathrm e}^{x}\right )\)	\(15\)
default	\(x +{\mathrm e}^{\frac {8}{3}} x +{\mathrm e}^{\frac {8}{3}} {\mathrm e}^{x}\)	\(16\)
norman	\(\left ({\mathrm e}^{\frac {8}{3}}+1\right ) x +{\mathrm e}^{\frac {8}{3}} {\mathrm e}^{x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(4/3)^2*exp(x)+exp(4/3)^2+1,x,method=_RETURNVERBOSE)

[Out]

x+exp(8/3)*x+exp(x+8/3)

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maxima [A]  time = 0.35, size = 10, normalized size = 0.59 \begin {gather*} x e^{\frac {8}{3}} + x + e^{\left (x + \frac {8}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(4/3)^2*exp(x)+exp(4/3)^2+1,x, algorithm="maxima")

[Out]

x*e^(8/3) + x + e^(x + 8/3)

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mupad [B]  time = 0.06, size = 11, normalized size = 0.65 \begin {gather*} {\mathrm {e}}^{x+\frac {8}{3}}+x\,\left ({\mathrm {e}}^{8/3}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(8/3) + exp(8/3)*exp(x) + 1,x)

[Out]

exp(x + 8/3) + x*(exp(8/3) + 1)

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sympy [A]  time = 0.09, size = 15, normalized size = 0.88 \begin {gather*} x \left (1 + e^{\frac {8}{3}}\right ) + e^{\frac {8}{3}} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(4/3)**2*exp(x)+exp(4/3)**2+1,x)

[Out]

x*(1 + exp(8/3)) + exp(8/3)*exp(x)

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