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 = 0.0147195, 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 \[ \frac{e^{(a+2) x}}{a+2} \]

Antiderivative was successfully verified.

[In]  Int[E^(2*x + a*x),x]

[Out]

E^((2 + a)*x)/(2 + a)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int e^{a x + 2 x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(a*x+2*x),x)

[Out]

Integral(exp(a*x + 2*x), x)

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Mathematica [A]  time = 0.00333198, 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 = 0.001, size = 15, normalized size = 1.2 \[{\frac{{{\rm e}^{ax+2\,x}}}{2+a}} \]

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 = 1.40839, size = 19, normalized size = 1.46 \[ \frac{e^{\left (a x + 2 \, x\right )}}{a + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(a*x + 2*x),x, algorithm="maxima")

[Out]

e^(a*x + 2*x)/(a + 2)

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Fricas [A]  time = 0.211677, size = 16, normalized size = 1.23 \[ \frac{e^{\left ({\left (a + 2\right )} x\right )}}{a + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(a*x + 2*x),x, algorithm="fricas")

[Out]

e^((a + 2)*x)/(a + 2)

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Sympy [A]  time = 0.071039, 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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GIAC/XCAS [A]  time = 0.199825, size = 19, normalized size = 1.46 \[ \frac{e^{\left (a x + 2 \, x\right )}}{a + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(a*x + 2*x),x, algorithm="giac")

[Out]

e^(a*x + 2*x)/(a + 2)