3.67.68 \(\int \frac {1}{4} (4+5 e^{-5 x/4}) \, dx\)

Optimal. Leaf size=14 \[ \frac {1}{e^3}-e^{-5 x/4}+x \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.79, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {12, 2194} \begin {gather*} x-e^{-5 x/4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + 5/E^((5*x)/4))/4,x]

[Out]

-E^((-5*x)/4) + x

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 {1}{4} \int \left (4+5 e^{-5 x/4}\right ) \, dx\\ &=x+\frac {5}{4} \int e^{-5 x/4} \, dx\\ &=-e^{-5 x/4}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 0.79 \begin {gather*} -e^{-5 x/4}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 5/E^((5*x)/4))/4,x]

[Out]

-E^((-5*x)/4) + x

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fricas [A]  time = 0.57, size = 8, normalized size = 0.57 \begin {gather*} x - e^{\left (-\frac {5}{4} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/4*exp(-5/4*x)+1,x, algorithm="fricas")

[Out]

x - e^(-5/4*x)

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giac [A]  time = 0.13, size = 8, normalized size = 0.57 \begin {gather*} x - e^{\left (-\frac {5}{4} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/4*exp(-5/4*x)+1,x, algorithm="giac")

[Out]

x - e^(-5/4*x)

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maple [A]  time = 0.02, size = 9, normalized size = 0.64

method	result	size
default	\(x -{\mathrm e}^{-\frac {5 x}{4}}\)	\(9\)
norman	\(x -{\mathrm e}^{-\frac {5 x}{4}}\)	\(9\)
risch	\(x -{\mathrm e}^{-\frac {5 x}{4}}\)	\(9\)
derivativedivides	\(-{\mathrm e}^{-\frac {5 x}{4}}-\frac {4 \ln \left ({\mathrm e}^{-\frac {5 x}{4}}\right )}{5}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5/4*exp(-5/4*x)+1,x,method=_RETURNVERBOSE)

[Out]

x-exp(-5/4*x)

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maxima [A]  time = 0.37, size = 8, normalized size = 0.57 \begin {gather*} x - e^{\left (-\frac {5}{4} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/4*exp(-5/4*x)+1,x, algorithm="maxima")

[Out]

x - e^(-5/4*x)

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mupad [B]  time = 0.05, size = 8, normalized size = 0.57 \begin {gather*} x-{\mathrm {e}}^{-\frac {5\,x}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*exp(-(5*x)/4))/4 + 1,x)

[Out]

x - exp(-(5*x)/4)

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sympy [A]  time = 0.07, size = 8, normalized size = 0.57 \begin {gather*} x - e^{- \frac {5 x}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5/4*exp(-5/4*x)+1,x)

[Out]

x - exp(-5*x/4)

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