3.504 \(\int (a^{k x}+a^{l x})^3 \, dx\)

Optimal. Leaf size=79 \[ \frac {3 a^{x (2 k+l)}}{\log (a) (2 k+l)}+\frac {3 a^{x (k+2 l)}}{\log (a) (k+2 l)}+\frac {a^{3 k x}}{3 k \log (a)}+\frac {a^{3 l x}}{3 l \log (a)} \]

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Rubi [A]  time = 0.10, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6742, 2194} \[ \frac {3 a^{x (2 k+l)}}{\log (a) (2 k+l)}+\frac {3 a^{x (k+2 l)}}{\log (a) (k+2 l)}+\frac {a^{3 k x}}{3 k \log (a)}+\frac {a^{3 l x}}{3 l \log (a)} \]

Antiderivative was successfully verified.

[In]

Int[(a^(k*x) + a^(l*x))^3,x]

[Out]

a^(3*k*x)/(3*k*Log[a]) + a^(3*l*x)/(3*l*Log[a]) + (3*a^((2*k + l)*x))/((2*k + l)*Log[a]) + (3*a^((k + 2*l)*x))
/((k + 2*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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \left (a^{k x}+a^{l x}\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \left (e^{k x}+e^{l x}\right )^3 \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {\operatorname {Subst}\left (\int \left (e^{3 k x}+e^{3 l x}+3 e^{(2 k+l) x}+3 e^{(k+2 l) x}\right ) \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {\operatorname {Subst}\left (\int e^{3 k x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {\operatorname {Subst}\left (\int e^{3 l x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {3 \operatorname {Subst}\left (\int e^{(2 k+l) x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {3 \operatorname {Subst}\left (\int e^{(k+2 l) x} \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {a^{3 k x}}{3 k \log (a)}+\frac {a^{3 l x}}{3 l \log (a)}+\frac {3 a^{(2 k+l) x}}{(2 k+l) \log (a)}+\frac {3 a^{(k+2 l) x}}{(k+2 l) \log (a)}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 65, normalized size = 0.82 \[ \frac {\frac {9 a^{x (2 k+l)}}{2 k+l}+\frac {9 a^{x (k+2 l)}}{k+2 l}+\frac {a^{3 k x}}{k}+\frac {a^{3 l x}}{l}}{3 \log (a)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^(k*x) + a^(l*x))^3,x]

[Out]

(a^(3*k*x)/k + a^(3*l*x)/l + (9*a^((2*k + l)*x))/(2*k + l) + (9*a^((k + 2*l)*x))/(k + 2*l))/(3*Log[a])

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a^{k x}+a^{l x}\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[(a^(k*x) + a^(l*x))^3,x]

[Out]

Could not integrate

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fricas [A]  time = 1.00, size = 130, normalized size = 1.65 \[ \frac {9 \, {\left (2 \, k^{2} l + k l^{2}\right )} a^{k x} a^{2 \, l x} + 9 \, {\left (k^{2} l + 2 \, k l^{2}\right )} a^{2 \, k x} a^{l x} + {\left (2 \, k^{2} l + 5 \, k l^{2} + 2 \, l^{3}\right )} a^{3 \, k x} + {\left (2 \, k^{3} + 5 \, k^{2} l + 2 \, k l^{2}\right )} a^{3 \, l x}}{3 \, {\left (2 \, k^{3} l + 5 \, k^{2} l^{2} + 2 \, k l^{3}\right )} \log \relax (a)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^(k*x)+a^(l*x))^3,x, algorithm="fricas")

[Out]

1/3*(9*(2*k^2*l + k*l^2)*a^(k*x)*a^(2*l*x) + 9*(k^2*l + 2*k*l^2)*a^(2*k*x)*a^(l*x) + (2*k^2*l + 5*k*l^2 + 2*l^
3)*a^(3*k*x) + (2*k^3 + 5*k^2*l + 2*k*l^2)*a^(3*l*x))/((2*k^3*l + 5*k^2*l^2 + 2*k*l^3)*log(a))

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giac [C]  time = 0.82, size = 1033, normalized size = 13.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^(k*x)+a^(l*x))^3,x, algorithm="giac")

[Out]

2/3*(2*k*cos(-3/2*pi*k*x*sgn(a) + 3/2*pi*k*x)*log(abs(a))/(4*k^2*log(abs(a))^2 + (pi*k*sgn(a) - pi*k)^2) - (pi
*k*sgn(a) - pi*k)*sin(-3/2*pi*k*x*sgn(a) + 3/2*pi*k*x)/(4*k^2*log(abs(a))^2 + (pi*k*sgn(a) - pi*k)^2))*abs(a)^
(3*k*x) + 2/3*(2*l*cos(-3/2*pi*l*x*sgn(a) + 3/2*pi*l*x)*log(abs(a))/(4*l^2*log(abs(a))^2 + (pi*l*sgn(a) - pi*l
)^2) - (pi*l*sgn(a) - pi*l)*sin(-3/2*pi*l*x*sgn(a) + 3/2*pi*l*x)/(4*l^2*log(abs(a))^2 + (pi*l*sgn(a) - pi*l)^2
))*abs(a)^(3*l*x) - 1/2*I*abs(a)^(3*k*x)*(-2*I*e^(3/2*I*pi*k*x*sgn(a) - 3/2*I*pi*k*x)/(3*I*pi*k*sgn(a) - 3*I*p
i*k + 6*k*log(abs(a))) + 2*I*e^(-3/2*I*pi*k*x*sgn(a) + 3/2*I*pi*k*x)/(-3*I*pi*k*sgn(a) + 3*I*pi*k + 6*k*log(ab
s(a)))) - 1/2*I*abs(a)^(3*l*x)*(-2*I*e^(3/2*I*pi*l*x*sgn(a) - 3/2*I*pi*l*x)/(3*I*pi*l*sgn(a) - 3*I*pi*l + 6*l*
log(abs(a))) + 2*I*e^(-3/2*I*pi*l*x*sgn(a) + 3/2*I*pi*l*x)/(-3*I*pi*l*sgn(a) + 3*I*pi*l + 6*l*log(abs(a)))) +
6*(2*(2*k*log(abs(a)) + l*log(abs(a)))*cos(-pi*k*x*sgn(a) - 1/2*pi*l*x*sgn(a) + pi*k*x + 1/2*pi*l*x)/((2*pi*k*
sgn(a) + pi*l*sgn(a) - 2*pi*k - pi*l)^2 + 4*(2*k*log(abs(a)) + l*log(abs(a)))^2) - (2*pi*k*sgn(a) + pi*l*sgn(a
) - 2*pi*k - pi*l)*sin(-pi*k*x*sgn(a) - 1/2*pi*l*x*sgn(a) + pi*k*x + 1/2*pi*l*x)/((2*pi*k*sgn(a) + pi*l*sgn(a)
- 2*pi*k - pi*l)^2 + 4*(2*k*log(abs(a)) + l*log(abs(a)))^2))*e^((2*k*log(abs(a)) + l*log(abs(a)))*x) - 1/2*I*
(-6*I*e^(I*pi*k*x*sgn(a) + 1/2*I*pi*l*x*sgn(a) - I*pi*k*x - 1/2*I*pi*l*x)/(2*I*pi*k*sgn(a) + I*pi*l*sgn(a) - 2
*I*pi*k - I*pi*l + 4*k*log(abs(a)) + 2*l*log(abs(a))) + 6*I*e^(-I*pi*k*x*sgn(a) - 1/2*I*pi*l*x*sgn(a) + I*pi*k
*x + 1/2*I*pi*l*x)/(-2*I*pi*k*sgn(a) - I*pi*l*sgn(a) + 2*I*pi*k + I*pi*l + 4*k*log(abs(a)) + 2*l*log(abs(a))))
*e^((2*k*log(abs(a)) + l*log(abs(a)))*x) + 6*(2*(k*log(abs(a)) + 2*l*log(abs(a)))*cos(-1/2*pi*k*x*sgn(a) - pi*
l*x*sgn(a) + 1/2*pi*k*x + pi*l*x)/((pi*k*sgn(a) + 2*pi*l*sgn(a) - pi*k - 2*pi*l)^2 + 4*(k*log(abs(a)) + 2*l*lo
g(abs(a)))^2) - (pi*k*sgn(a) + 2*pi*l*sgn(a) - pi*k - 2*pi*l)*sin(-1/2*pi*k*x*sgn(a) - pi*l*x*sgn(a) + 1/2*pi*
k*x + pi*l*x)/((pi*k*sgn(a) + 2*pi*l*sgn(a) - pi*k - 2*pi*l)^2 + 4*(k*log(abs(a)) + 2*l*log(abs(a)))^2))*e^((k
*log(abs(a)) + 2*l*log(abs(a)))*x) - 1/2*I*(-6*I*e^(1/2*I*pi*k*x*sgn(a) + I*pi*l*x*sgn(a) - 1/2*I*pi*k*x - I*p
i*l*x)/(I*pi*k*sgn(a) + 2*I*pi*l*sgn(a) - I*pi*k - 2*I*pi*l + 2*k*log(abs(a)) + 4*l*log(abs(a))) + 6*I*e^(-1/2
*I*pi*k*x*sgn(a) - I*pi*l*x*sgn(a) + 1/2*I*pi*k*x + I*pi*l*x)/(-I*pi*k*sgn(a) - 2*I*pi*l*sgn(a) + I*pi*k + 2*I
*pi*l + 2*k*log(abs(a)) + 4*l*log(abs(a))))*e^((k*log(abs(a)) + 2*l*log(abs(a)))*x)

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maple [A]  time = 0.07, size = 84, normalized size = 1.06

method	result	size
risch	\(\frac {a^{3 k x}}{3 k \ln \relax (a )}+\frac {a^{3 l x}}{3 l \ln \relax (a )}+\frac {3 a^{k x} a^{2 l x}}{\ln \relax (a ) \left (k +2 l \right )}+\frac {3 a^{2 k x} a^{l x}}{\ln \relax (a ) \left (2 k +l \right )}\)	\(84\)
norman	\(\frac {{\mathrm e}^{3 k x \ln \relax (a )}}{3 k \ln \relax (a )}+\frac {{\mathrm e}^{3 l x \ln \relax (a )}}{3 l \ln \relax (a )}+\frac {3 \,{\mathrm e}^{k x \ln \relax (a )} {\mathrm e}^{2 l x \ln \relax (a )}}{\ln \relax (a ) \left (k +2 l \right )}+\frac {3 \,{\mathrm e}^{2 k x \ln \relax (a )} {\mathrm e}^{l x \ln \relax (a )}}{\ln \relax (a ) \left (2 k +l \right )}\)	\(90\)
meijerg	error in int/gbinthm/express: improper op or subscript selector\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^(k*x)+a^(l*x))^3,x,method=_RETURNVERBOSE)

[Out]

1/3/k/ln(a)*(a^(k*x))^3+1/3/l/ln(a)*(a^(l*x))^3+3/ln(a)/(k+2*l)*a^(k*x)*(a^(l*x))^2+3/ln(a)/(2*k+l)*(a^(k*x))^
2*a^(l*x)

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maxima [A]  time = 0.55, size = 77, normalized size = 0.97 \[ \frac {3 \, a^{2 \, k x + l x}}{{\left (2 \, k + l\right )} \log \relax (a)} + \frac {3 \, a^{k x + 2 \, l x}}{{\left (k + 2 \, l\right )} \log \relax (a)} + \frac {a^{3 \, k x}}{3 \, k \log \relax (a)} + \frac {a^{3 \, l x}}{3 \, l \log \relax (a)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^(k*x)+a^(l*x))^3,x, algorithm="maxima")

[Out]

3*a^(2*k*x + l*x)/((2*k + l)*log(a)) + 3*a^(k*x + 2*l*x)/((k + 2*l)*log(a)) + 1/3*a^(3*k*x)/(k*log(a)) + 1/3*a
^(3*l*x)/(l*log(a))

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mupad [B]  time = 0.39, size = 81, normalized size = 1.03 \[ \frac {3\,a^{k\,x}\,a^{2\,l\,x}}{k\,\ln \relax (a)+2\,l\,\ln \relax (a)}+\frac {3\,a^{2\,k\,x}\,a^{l\,x}}{2\,k\,\ln \relax (a)+l\,\ln \relax (a)}+\frac {a^{3\,k\,x}}{3\,k\,\ln \relax (a)}+\frac {a^{3\,l\,x}}{3\,l\,\ln \relax (a)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^(k*x) + a^(l*x))^3,x)

[Out]

(3*a^(k*x)*a^(2*l*x))/(k*log(a) + 2*l*log(a)) + (3*a^(2*k*x)*a^(l*x))/(2*k*log(a) + l*log(a)) + a^(3*k*x)/(3*k
*log(a)) + a^(3*l*x)/(3*l*log(a))

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sympy [A]  time = 23.44, size = 665, normalized size = 8.42 \[ \begin {cases} 8 x & \text {for}\: a = 1 \wedge \left (a = 1 \vee k = 0\right ) \wedge \left (a = 1 \vee l = 0\right ) \\\frac {a^{3 l x}}{3 l \log {\relax (a )}} + \frac {3 a^{2 l x}}{2 l \log {\relax (a )}} + \frac {3 a^{l x}}{l \log {\relax (a )}} + x & \text {for}\: k = 0 \\\frac {a^{3 l x}}{3 l \log {\relax (a )}} + 3 x - \frac {a^{- 3 l x}}{l \log {\relax (a )}} - \frac {a^{- 6 l x}}{6 l \log {\relax (a )}} & \text {for}\: k = - 2 l \\\frac {2 a^{\frac {3 l x}{2}}}{l \log {\relax (a )}} + \frac {a^{3 l x}}{3 l \log {\relax (a )}} + 3 x - \frac {2 a^{- \frac {3 l x}{2}}}{3 l \log {\relax (a )}} & \text {for}\: k = - \frac {l}{2} \\\frac {a^{3 k x}}{3 k \log {\relax (a )}} + \frac {3 a^{2 k x}}{2 k \log {\relax (a )}} + \frac {3 a^{k x}}{k \log {\relax (a )}} + x & \text {for}\: l = 0 \\\frac {2 a^{3 k x} k^{2} l}{6 k^{3} l \log {\relax (a )} + 15 k^{2} l^{2} \log {\relax (a )} + 6 k l^{3} \log {\relax (a )}} + \frac {5 a^{3 k x} k l^{2}}{6 k^{3} l \log {\relax (a )} + 15 k^{2} l^{2} \log {\relax (a )} + 6 k l^{3} \log {\relax (a )}} + \frac {2 a^{3 k x} l^{3}}{6 k^{3} l \log {\relax (a )} + 15 k^{2} l^{2} \log {\relax (a )} + 6 k l^{3} \log {\relax (a )}} + \frac {9 a^{2 k x} a^{l x} k^{2} l}{6 k^{3} l \log {\relax (a )} + 15 k^{2} l^{2} \log {\relax (a )} + 6 k l^{3} \log {\relax (a )}} + \frac {18 a^{2 k x} a^{l x} k l^{2}}{6 k^{3} l \log {\relax (a )} + 15 k^{2} l^{2} \log {\relax (a )} + 6 k l^{3} \log {\relax (a )}} + \frac {18 a^{k x} a^{2 l x} k^{2} l}{6 k^{3} l \log {\relax (a )} + 15 k^{2} l^{2} \log {\relax (a )} + 6 k l^{3} \log {\relax (a )}} + \frac {9 a^{k x} a^{2 l x} k l^{2}}{6 k^{3} l \log {\relax (a )} + 15 k^{2} l^{2} \log {\relax (a )} + 6 k l^{3} \log {\relax (a )}} + \frac {2 a^{3 l x} k^{3}}{6 k^{3} l \log {\relax (a )} + 15 k^{2} l^{2} \log {\relax (a )} + 6 k l^{3} \log {\relax (a )}} + \frac {5 a^{3 l x} k^{2} l}{6 k^{3} l \log {\relax (a )} + 15 k^{2} l^{2} \log {\relax (a )} + 6 k l^{3} \log {\relax (a )}} + \frac {2 a^{3 l x} k l^{2}}{6 k^{3} l \log {\relax (a )} + 15 k^{2} l^{2} \log {\relax (a )} + 6 k l^{3} \log {\relax (a )}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**(k*x)+a**(l*x))**3,x)

[Out]

Piecewise((8*x, Eq(a, 1) & (Eq(a, 1) | Eq(k, 0)) & (Eq(a, 1) | Eq(l, 0))), (a**(3*l*x)/(3*l*log(a)) + 3*a**(2*
l*x)/(2*l*log(a)) + 3*a**(l*x)/(l*log(a)) + x, Eq(k, 0)), (a**(3*l*x)/(3*l*log(a)) + 3*x - a**(-3*l*x)/(l*log(
a)) - a**(-6*l*x)/(6*l*log(a)), Eq(k, -2*l)), (2*a**(3*l*x/2)/(l*log(a)) + a**(3*l*x)/(3*l*log(a)) + 3*x - 2*a
**(-3*l*x/2)/(3*l*log(a)), Eq(k, -l/2)), (a**(3*k*x)/(3*k*log(a)) + 3*a**(2*k*x)/(2*k*log(a)) + 3*a**(k*x)/(k*
log(a)) + x, Eq(l, 0)), (2*a**(3*k*x)*k**2*l/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*log(a)) + 5*a**
(3*k*x)*k*l**2/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*log(a)) + 2*a**(3*k*x)*l**3/(6*k**3*l*log(a)
+ 15*k**2*l**2*log(a) + 6*k*l**3*log(a)) + 9*a**(2*k*x)*a**(l*x)*k**2*l/(6*k**3*l*log(a) + 15*k**2*l**2*log(a)
+ 6*k*l**3*log(a)) + 18*a**(2*k*x)*a**(l*x)*k*l**2/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*log(a))
+ 18*a**(k*x)*a**(2*l*x)*k**2*l/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*log(a)) + 9*a**(k*x)*a**(2*l
*x)*k*l**2/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*log(a)) + 2*a**(3*l*x)*k**3/(6*k**3*l*log(a) + 15
*k**2*l**2*log(a) + 6*k*l**3*log(a)) + 5*a**(3*l*x)*k**2*l/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*l
og(a)) + 2*a**(3*l*x)*k*l**2/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*log(a)), True))

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