3.156 \(\int e^{2 x+a x} \, dx\)

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

[Out]

E^((2 + a)*x)/(2 + a)

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Rubi [A]  time = 0.0075806, antiderivative size = 13, 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 = {2227, 2194} \[ \frac{e^{(a+2) x}}{a+2} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*x + a*x),x]

[Out]

E^((2 + a)*x)/(2 + a)

Rule 2227

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]

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

Mathematica [A]  time = 0.0033436, size = 13, normalized size = 1. \[ \frac{e^{(a+2) x}}{a+2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x + a*x),x]

[Out]

E^((2 + a)*x)/(2 + a)

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Maple [A]  time = 0.002, size = 15, normalized size = 1.2 \begin{align*}{\frac{{{\rm e}^{ax+2\,x}}}{2+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.933516, size = 19, normalized size = 1.46 \begin{align*} \frac{e^{\left (a x + 2 \, x\right )}}{a + 2} \end{align*}

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 = 1.70432, size = 31, normalized size = 2.38 \begin{align*} \frac{e^{\left ({\left (a + 2\right )} x\right )}}{a + 2} \end{align*}

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 = 0.103013, size = 14, normalized size = 1.08 \begin{align*} \begin{cases} \frac{e^{a x + 2 x}}{a + 2} & \text{for}\: a + 2 \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}

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, 0)), (x, True))

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Giac [A]  time = 1.08114, size = 19, normalized size = 1.46 \begin{align*} \frac{e^{\left (a x + 2 \, x\right )}}{a + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(a*x + 2*x)/(a + 2)