3.6.2 \(\int (a^{k x}+a^{l x}) \, dx\) [502]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {2225} \begin {gather*} \frac {a^{k x}}{k \log (a)}+\frac {a^{l x}}{l \log (a)} \end {gather*}

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 2225

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*}

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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} \frac {a^{k x}}{k \log (a)}+\frac {a^{l x}}{l \log (a)} \end {gather*}

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 = 0.02, size = 28, normalized size = 1.04

method	result	size
default	\(\frac {a^{k x}}{k \ln \left (a \right )}+\frac {a^{l x}}{l \ln \left (a \right )}\)	\(28\)
risch	\(\frac {a^{k x}}{k \ln \left (a \right )}+\frac {a^{l x}}{l \ln \left (a \right )}\)	\(28\)
norman	\(\frac {{\mathrm e}^{k x \ln \left (a \right )}}{k \ln \left (a \right )}+\frac {{\mathrm e}^{l x \ln \left (a \right )}}{l \ln \left (a \right )}\)	\(30\)
meijerg	\(-\frac {1-{\mathrm e}^{k x \ln \left (a \right )}}{k \ln \left (a \right )}-\frac {1-{\mathrm e}^{l x \ln \left (a \right )}}{l \ln \left (a \right )}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^(k*x)+a^(l*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 2.09, size = 27, normalized size = 1.00 \begin {gather*} \frac {a^{k x}}{k \log \left (a\right )} + \frac {a^{l x}}{l \log \left (a\right )} \end {gather*}

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 = 0.49, size = 26, normalized size = 0.96 \begin {gather*} \frac {a^{l x} k + a^{k x} l}{k l \log \left (a\right )} \end {gather*}

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 = 0.05, size = 29, normalized size = 1.07 \begin {gather*} \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 {gather*}

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 [A]
time = 1.04, size = 27, normalized size = 1.00 \begin {gather*} \frac {a^{k x}}{k \log \left (a\right )} + \frac {a^{l x}}{l \log \left (a\right )} \end {gather*}

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))

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Mupad [B]
time = 0.36, size = 26, normalized size = 0.96 \begin {gather*} \frac {a^{k\,x}\,l+a^{l\,x}\,k}{k\,l\,\ln \left (a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^(k*x) + a^(l*x),x)

[Out]

(a^(k*x)*l + a^(l*x)*k)/(k*l*log(a))

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