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

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

[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 = {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 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]

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]

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 = 0.00, size = 13, normalized size = 1.00 \[ \frac {e^{(a+2) x}}{a+2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x + a*x),x]

[Out]

E^((2 + a)*x)/(2 + a)

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fricas [A]  time = 0.40, size = 12, normalized size = 0.92 \[ \frac {e^{\left ({\left (a + 2\right )} x\right )}}{a + 2} \]

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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giac [A]  time = 1.12, size = 14, normalized size = 1.08 \[ \frac {e^{\left (a x + 2 \, x\right )}}{a + 2} \]

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)

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maple [A]  time = 0.00, size = 15, normalized size = 1.15 \[ \frac {{\mathrm e}^{a x +2 x}}{a +2} \]

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 = 0.42, size = 14, normalized size = 1.08 \[ \frac {e^{\left (a x + 2 \, x\right )}}{a + 2} \]

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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mupad [B]  time = 0.05, size = 14, normalized size = 1.08 \[ \frac {{\mathrm {e}}^{2\,x+a\,x}}{a+2} \]

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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sympy [A]  time = 0.09, size = 14, normalized size = 1.08 \[ \begin {cases} \frac {e^{a x + 2 x}}{a + 2} & \text {for}\: a + 2 \neq 0 \\x & \text {otherwise} \end {cases} \]

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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