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

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

[Out]

exp(a*x)/a

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \[ \frac {e^{a x}}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(a*x)/a

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fricas [A]  time = 0.40, size = 8, normalized size = 0.89 \[ \frac {e^{\left (a x\right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(a*x)/a

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giac [A]  time = 1.04, size = 8, normalized size = 0.89 \[ \frac {e^{\left (a x\right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(a*x)/a

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maple [A]  time = 0.00, size = 9, normalized size = 1.00 \[ \frac {{\mathrm e}^{a x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a*x),x)

[Out]

exp(a*x)/a

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maxima [A]  time = 0.41, size = 8, normalized size = 0.89 \[ \frac {e^{\left (a x\right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(a*x)/a

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mupad [B]  time = 0.03, size = 8, normalized size = 0.89 \[ \frac {{\mathrm {e}}^{a\,x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a*x),x)

[Out]

exp(a*x)/a

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sympy [A]  time = 0.05, size = 7, normalized size = 0.78 \[ \begin {cases} \frac {e^{a x}}{a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \]

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))

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