3.73.9 \(\int \frac {8}{15} e^{4 x/3} \, dx\)

Optimal. Leaf size=11 \[ \frac {2}{5} e^{4 x/3} \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2194} \begin {gather*} \frac {2}{5} e^{4 x/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8*E^((4*x)/3))/15,x]

[Out]

(2*E^((4*x)/3))/5

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {8}{15} \int e^{4 x/3} \, dx\\ &=\frac {2}{5} e^{4 x/3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \frac {2}{5} e^{4 x/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*E^((4*x)/3))/15,x]

[Out]

(2*E^((4*x)/3))/5

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fricas [A]  time = 1.14, size = 6, normalized size = 0.55 \begin {gather*} \frac {2}{5} \, e^{\left (\frac {4}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8/15*exp(4/3*x),x, algorithm="fricas")

[Out]

2/5*e^(4/3*x)

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giac [A]  time = 0.22, size = 6, normalized size = 0.55 \begin {gather*} \frac {2}{5} \, e^{\left (\frac {4}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8/15*exp(4/3*x),x, algorithm="giac")

[Out]

2/5*e^(4/3*x)

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maple [A]  time = 0.02, size = 7, normalized size = 0.64

method	result	size
gosper	\(\frac {2 \,{\mathrm e}^{\frac {4 x}{3}}}{5}\)	\(7\)
derivativedivides	\(\frac {2 \,{\mathrm e}^{\frac {4 x}{3}}}{5}\)	\(7\)
default	\(\frac {2 \,{\mathrm e}^{\frac {4 x}{3}}}{5}\)	\(7\)
norman	\(\frac {2 \,{\mathrm e}^{\frac {4 x}{3}}}{5}\)	\(7\)
risch	\(\frac {2 \,{\mathrm e}^{\frac {4 x}{3}}}{5}\)	\(7\)
meijerg	\(-\frac {2}{5}+\frac {2 \,{\mathrm e}^{\frac {4 x}{3}}}{5}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8/15*exp(4/3*x),x,method=_RETURNVERBOSE)

[Out]

2/5*exp(4/3*x)

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maxima [A]  time = 0.35, size = 6, normalized size = 0.55 \begin {gather*} \frac {2}{5} \, e^{\left (\frac {4}{3} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8/15*exp(4/3*x),x, algorithm="maxima")

[Out]

2/5*e^(4/3*x)

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mupad [B]  time = 0.02, size = 6, normalized size = 0.55 \begin {gather*} \frac {2\,{\mathrm {e}}^{\frac {4\,x}{3}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*exp((4*x)/3))/15,x)

[Out]

(2*exp((4*x)/3))/5

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sympy [A]  time = 0.05, size = 8, normalized size = 0.73 \begin {gather*} \frac {2 e^{\frac {4 x}{3}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(8/15*exp(4/3*x),x)

[Out]

2*exp(4*x/3)/5

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