3.152 \(\int e^{a x} \, dx\)

Optimal. Leaf size=9 \[ \frac{e^{a x}}{a} \]

[Out]

E^(a*x)/a

________________________________________________________________________________________

Rubi [A]  time = 0.0017289, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2194} \[ \frac{e^{a x}}{a} \]

Antiderivative was successfully verified.

[In]

Int[E^(a*x),x]

[Out]

E^(a*x)/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]

Rubi steps

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

Mathematica [A]  time = 0.0011267, size = 9, normalized size = 1. \[ \frac{e^{a x}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(a*x),x]

[Out]

E^(a*x)/a

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 9, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{ax}}}{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a*x),x)

[Out]

exp(a*x)/a

________________________________________________________________________________________

Maxima [A]  time = 0.955901, size = 11, normalized size = 1.22 \begin{align*} \frac{e^{\left (a x\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(a*x)/a

________________________________________________________________________________________

Fricas [A]  time = 1.56077, size = 15, normalized size = 1.67 \begin{align*} \frac{e^{\left (a x\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(a*x)/a

________________________________________________________________________________________

Sympy [A]  time = 0.054832, size = 7, normalized size = 0.78 \begin{align*} \begin{cases} \frac{e^{a x}}{a} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a*x),x)

[Out]

Piecewise((exp(a*x)/a, Ne(a, 0)), (x, True))

________________________________________________________________________________________

Giac [A]  time = 1.07132, size = 11, normalized size = 1.22 \begin{align*} \frac{e^{\left (a x\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(a*x)/a