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

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])

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Rubi [A]  time = 0.0243129, 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 \[ \frac{a^{k x}}{k \log (a)}+\frac{a^{l x}}{l \log (a)} \]

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])

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Rubi in Sympy [A]  time = 1.02179, size = 19, normalized size = 0.7 \[ \frac{a^{k x}}{k \log{\left (a \right )}} + \frac{a^{l x}}{l \log{\left (a \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(a**(k*x)+a**(l*x),x)

[Out]

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

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Mathematica [A]  time = 0.00928911, 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])

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Maple [A]  time = 0.004, size = 28, normalized size = 1. \[{\frac{{a}^{kx}}{k\ln \left ( a \right ) }}+{\frac{{a}^{lx}}{l\ln \left ( a \right ) }} \]

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)

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Maxima [A]  time = 1.3419, size = 36, normalized size = 1.33 \[ \frac{a^{k x}}{k \log \left (a\right )} + \frac{a^{l x}}{l \log \left (a\right )} \]

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))

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Fricas [A]  time = 0.214634, size = 35, normalized size = 1.3 \[ \frac{a^{l x} k + a^{k x} l}{k l \log \left (a\right )} \]

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))

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Sympy [A]  time = 0.566064, size = 29, normalized size = 1.07 \[ \begin{cases} \frac{a^{k x}}{k \log{\left (a \right )}} & \text{for}\: k \log{\left (a \right )} \neq 0 \\x & \text{otherwise} \end{cases} + \begin{cases} \frac{a^{l x}}{l \log{\left (a \right )}} & \text{for}\: l \log{\left (a \right )} \neq 0 \\x & \text{otherwise} \end{cases} \]

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))

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GIAC/XCAS [A]  time = 0.195233, size = 36, normalized size = 1.33 \[ \frac{a^{k x}}{k{\rm ln}\left (a\right )} + \frac{a^{l x}}{l{\rm ln}\left (a\right )} \]

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*ln(a)) + a^(l*x)/(l*ln(a))