3.2.56 \(\int e^{2 x+a x} \, dx\) [156]

Optimal. Leaf size=13 \[ \frac {e^{(2+a) x}}{2+a} \]

[Out]

exp((2+a)*x)/(2+a)

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Rubi [A]
time = 0.01, antiderivative size = 13, 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 = {2259, 2225} \begin {gather*} \frac {e^{(a+2) x}}{a+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*x + a*x),x]

[Out]

E^((2 + a)*x)/(2 + a)

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]

Rule 2259

Int[(u_.)*(F_)^((a_.) + (b_.)*(v_)), x_Symbol] :> Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F
, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] &&  !PowerOfLinearMatchQ[v, x]

Rubi steps

\begin {align*} \int e^{2 x+a x} \, dx &=\int e^{(2+a) x} \, dx\\ &=\frac {e^{(2+a) x}}{2+a}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {e^{(2+a) x}}{2+a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x + a*x),x]

[Out]

E^((2 + a)*x)/(2 + a)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.80, size = 20, normalized size = 1.54 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {E^{x \left (2+a\right )}}{2+a},a\text {!=}-2\right \}\right \},x\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{E ^ (x (2 + a)) / (2 + a), a != -2}}, x]

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Maple [A]
time = 0.01, size = 15, normalized size = 1.15

method result size
risch \(\frac {{\mathrm e}^{\left (2+a \right ) x}}{2+a}\) \(13\)
gosper \(\frac {{\mathrm e}^{a x +2 x}}{2+a}\) \(15\)
derivativedivides \(\frac {{\mathrm e}^{a x +2 x}}{2+a}\) \(15\)
default \(\frac {{\mathrm e}^{a x +2 x}}{2+a}\) \(15\)
norman \(\frac {{\mathrm e}^{a x +2 x}}{2+a}\) \(15\)
meijerg \(\frac {1-{\mathrm e}^{-x \left (-a -2\right )}}{-a -2}\) \(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a*x+2*x),x,method=_RETURNVERBOSE)

[Out]

1/(2+a)*exp(a*x+2*x)

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Maxima [A]
time = 0.26, size = 14, normalized size = 1.08 \begin {gather*} \frac {e^{\left (a x + 2 \, x\right )}}{a + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a*x+2*x),x, algorithm="maxima")

[Out]

e^(a*x + 2*x)/(a + 2)

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Fricas [A]
time = 0.30, size = 12, normalized size = 0.92 \begin {gather*} \frac {e^{\left ({\left (a + 2\right )} x\right )}}{a + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a*x+2*x),x, algorithm="fricas")

[Out]

e^((a + 2)*x)/(a + 2)

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Sympy [A]
time = 0.04, size = 14, normalized size = 1.08 \begin {gather*} \begin {cases} \frac {e^{a x + 2 x}}{a + 2} & \text {for}\: a \neq -2 \\x & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a*x+2*x),x)

[Out]

Piecewise((exp(a*x + 2*x)/(a + 2), Ne(a, -2)), (x, True))

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Giac [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {\mathrm {e}^{a x+2 x}}{a+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a*x+2*x),x)

[Out]

e^(a*x + 2*x)/(a + 2)

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Mupad [B]
time = 0.05, size = 14, normalized size = 1.08 \begin {gather*} \frac {{\mathrm {e}}^{2\,x+a\,x}}{a+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x + a*x),x)

[Out]

exp(2*x + a*x)/(a + 2)

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