3.22 \(\int 2 e^{2 x} y z \, dx\)

Optimal. Leaf size=8 \[ e^{2 x} y z \]

[Out]

E^(2*x)*y*z

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Rubi [A]  time = 0.0032724, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {12, 2194} \[ e^{2 x} y z \]

Antiderivative was successfully verified.

[In]

Int[2*E^(2*x)*y*z,x]

[Out]

E^(2*x)*y*z

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{align*} \int 2 e^{2 x} y z \, dx &=(2 y z) \int e^{2 x} \, dx\\ &=e^{2 x} y z\\ \end{align*}

Mathematica [A]  time = 0.0005961, size = 8, normalized size = 1. \[ e^{2 x} y z \]

Antiderivative was successfully verified.

[In]

Integrate[2*E^(2*x)*y*z,x]

[Out]

E^(2*x)*y*z

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Maple [A]  time = 0.002, size = 8, normalized size = 1. \begin{align*}{{\rm e}^{2\,x}}yz \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*exp(2*x)*y*z,x)

[Out]

exp(2*x)*y*z

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Maxima [A]  time = 0.932509, size = 9, normalized size = 1.12 \begin{align*} y z e^{\left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(2*x)*y*z,x, algorithm="maxima")

[Out]

y*z*e^(2*x)

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Fricas [A]  time = 1.86235, size = 18, normalized size = 2.25 \begin{align*} y z e^{\left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(2*x)*y*z,x, algorithm="fricas")

[Out]

y*z*e^(2*x)

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Sympy [A]  time = 0.06015, size = 7, normalized size = 0.88 \begin{align*} y z e^{2 x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(2*x)*y*z,x)

[Out]

y*z*exp(2*x)

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Giac [A]  time = 1.07112, size = 9, normalized size = 1.12 \begin{align*} y z e^{\left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*exp(2*x)*y*z,x, algorithm="giac")

[Out]

y*z*e^(2*x)