3.502 \(\int (a^{k x}+a^{l x}) \, dx\)

Optimal. Leaf size=27 \[ \frac{a^{k x}}{k \log (a)}+\frac{a^{l x}}{l \log (a)} \]

[Out]

a^(k*x)/(k*Log[a]) + a^(l*x)/(l*Log[a])

________________________________________________________________________________________

Rubi [A]  time = 0.010812, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2194} \[ \frac{a^{k x}}{k \log (a)}+\frac{a^{l x}}{l \log (a)} \]

Antiderivative was successfully verified.

[In]

Int[a^(k*x) + a^(l*x),x]

[Out]

a^(k*x)/(k*Log[a]) + a^(l*x)/(l*Log[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 \left (a^{k x}+a^{l x}\right ) \, dx &=\int a^{k x} \, dx+\int a^{l x} \, dx\\ &=\frac{a^{k x}}{k \log (a)}+\frac{a^{l x}}{l \log (a)}\\ \end{align*}

Mathematica [A]  time = 0.0058437, size = 27, normalized size = 1. \[ \frac{a^{k x}}{k \log (a)}+\frac{a^{l x}}{l \log (a)} \]

Antiderivative was successfully verified.

[In]

Integrate[a^(k*x) + a^(l*x),x]

[Out]

a^(k*x)/(k*Log[a]) + a^(l*x)/(l*Log[a])

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 28, normalized size = 1. \begin{align*}{\frac{{a}^{kx}}{k\ln \left ( a \right ) }}+{\frac{{a}^{lx}}{l\ln \left ( a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^(k*x)+a^(l*x),x)

[Out]

a^(k*x)/k/ln(a)+a^(l*x)/l/ln(a)

________________________________________________________________________________________

Maxima [A]  time = 0.948154, size = 36, normalized size = 1.33 \begin{align*} \frac{a^{k x}}{k \log \left (a\right )} + \frac{a^{l x}}{l \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^(k*x)+a^(l*x),x, algorithm="maxima")

[Out]

a^(k*x)/(k*log(a)) + a^(l*x)/(l*log(a))

________________________________________________________________________________________

Fricas [A]  time = 1.76971, size = 51, normalized size = 1.89 \begin{align*} \frac{a^{l x} k + a^{k x} l}{k l \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^(k*x)+a^(l*x),x, algorithm="fricas")

[Out]

(a^(l*x)*k + a^(k*x)*l)/(k*l*log(a))

________________________________________________________________________________________

Sympy [A]  time = 0.286201, size = 29, normalized size = 1.07 \begin{align*} \begin{cases} \frac{a^{k x}}{k \log{\left (a \right )}} & \text{for}\: k \log{\left (a \right )} \neq 0 \\x & \text{otherwise} \end{cases} + \begin{cases} \frac{a^{l x}}{l \log{\left (a \right )}} & \text{for}\: l \log{\left (a \right )} \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a**(k*x)+a**(l*x),x)

[Out]

Piecewise((a**(k*x)/(k*log(a)), Ne(k*log(a), 0)), (x, True)) + Piecewise((a**(l*x)/(l*log(a)), Ne(l*log(a), 0)
), (x, True))

________________________________________________________________________________________

Giac [A]  time = 1.06494, size = 36, normalized size = 1.33 \begin{align*} \frac{a^{k x}}{k \log \left (a\right )} + \frac{a^{l x}}{l \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^(k*x)+a^(l*x),x, algorithm="giac")

[Out]

a^(k*x)/(k*log(a)) + a^(l*x)/(l*log(a))