3.94.24 \(\int \frac {1}{9} (-25+18 e^{2 x}) \, dx\)

Optimal. Leaf size=12 \[ -6+e^{2 x}-\frac {25 x}{9} \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {12, 2194} \begin {gather*} e^{2 x}-\frac {25 x}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-25 + 18*E^(2*x))/9,x]

[Out]

E^(2*x) - (25*x)/9

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}{9} \int \left (-25+18 e^{2 x}\right ) \, dx\\ &=-\frac {25 x}{9}+2 \int e^{2 x} \, dx\\ &=e^{2 x}-\frac {25 x}{9}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.92 \begin {gather*} e^{2 x}-\frac {25 x}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25 + 18*E^(2*x))/9,x]

[Out]

E^(2*x) - (25*x)/9

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fricas [A]  time = 0.70, size = 8, normalized size = 0.67 \begin {gather*} -\frac {25}{9} \, x + e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(2*x)-25/9,x, algorithm="fricas")

[Out]

-25/9*x + e^(2*x)

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giac [A]  time = 0.18, size = 8, normalized size = 0.67 \begin {gather*} -\frac {25}{9} \, x + e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(2*x)-25/9,x, algorithm="giac")

[Out]

-25/9*x + e^(2*x)

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maple [A]  time = 0.02, size = 9, normalized size = 0.75

method	result	size
default	\(-\frac {25 x}{9}+{\mathrm e}^{2 x}\)	\(9\)
norman	\(-\frac {25 x}{9}+{\mathrm e}^{2 x}\)	\(9\)
risch	\(-\frac {25 x}{9}+{\mathrm e}^{2 x}\)	\(9\)
derivativedivides	\({\mathrm e}^{2 x}-\frac {25 \ln \left ({\mathrm e}^{2 x}\right )}{18}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*exp(2*x)-25/9,x,method=_RETURNVERBOSE)

[Out]

-25/9*x+exp(2*x)

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maxima [A]  time = 0.41, size = 8, normalized size = 0.67 \begin {gather*} -\frac {25}{9} \, x + e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(2*x)-25/9,x, algorithm="maxima")

[Out]

-25/9*x + e^(2*x)

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mupad [B]  time = 0.06, size = 8, normalized size = 0.67 \begin {gather*} {\mathrm {e}}^{2\,x}-\frac {25\,x}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*exp(2*x) - 25/9,x)

[Out]

exp(2*x) - (25*x)/9

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sympy [A]  time = 0.07, size = 8, normalized size = 0.67 \begin {gather*} - \frac {25 x}{9} + e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(2*x)-25/9,x)

[Out]

-25*x/9 + exp(2*x)

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