3.1.22 \(\int 2 e^{2 x} y z \, dx\) [22]

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

[Out]

exp(2*x)*y*z

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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, 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, 2225} \begin {gather*} e^{2 x} y z \end {gather*}

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 2225

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*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} e^{2 x} y z \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2*x)*y*z

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Mathics [A]
time = 1.69, size = 8, normalized size = 1.00 \begin {gather*} y z E^{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

y z E ^ (2 x)

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Maple [A]
time = 0.01, size = 8, normalized size = 1.00

method	result	size
gosper	\({\mathrm e}^{2 x} y z\)	\(8\)
derivativedivides	\({\mathrm e}^{2 x} y z\)	\(8\)
default	\({\mathrm e}^{2 x} y z\)	\(8\)
norman	\({\mathrm e}^{2 x} y z\)	\(8\)
risch	\({\mathrm e}^{2 x} y z\)	\(8\)
meijerg	\(-y z \left (1-{\mathrm e}^{2 x}\right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*exp(2*x)*y*z,x,method=_RETURNVERBOSE)

[Out]

exp(2*x)*y*z

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Maxima [A]
time = 0.25, size = 7, normalized size = 0.88 \begin {gather*} y z e^{\left (2 \, x\right )} \end {gather*}

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 = 0.32, size = 7, normalized size = 0.88 \begin {gather*} y z e^{\left (2 \, x\right )} \end {gather*}

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 = 0.03, size = 7, normalized size = 0.88 \begin {gather*} y z e^{2 x} \end {gather*}

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 = 0.00, size = 10, normalized size = 1.25 \begin {gather*} \frac {2}{2} y z \mathrm {e}^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(2*x)*y*z

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Mupad [B]
time = 0.02, size = 7, normalized size = 0.88 \begin {gather*} y\,z\,{\mathrm {e}}^{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

y*z*exp(2*x)

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