3.509 \(\int \left (a^{k x}-a^{l x}\right )^3 \, dx\)

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

[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])

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Rubi [A]  time = 0.187399, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{3 a^{x (2 k+l)}}{\log (a) (2 k+l)}+\frac{3 a^{x (k+2 l)}}{\log (a) (k+2 l)}+\frac{a^{3 k x}}{3 k \log (a)}-\frac{a^{3 l x}}{3 l \log (a)} \]

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])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a^{k x} - a^{l x}\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a**(k*x)-a**(l*x))**3,x)

[Out]

Integral((a**(k*x) - a**(l*x))**3, x)

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Mathematica [A]  time = 0.133242, size = 66, normalized size = 0.84 \[ \frac{-\frac{9 a^{x (2 k+l)}}{2 k+l}+\frac{9 a^{x (k+2 l)}}{k+2 l}+\frac{a^{3 k x}}{k}-\frac{a^{3 l x}}{l}}{3 \log (a)} \]

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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Maple [A]  time = 0.04, size = 90, normalized size = 1.1 \[{\frac{ \left ({{\rm e}^{kx\ln \left ( a \right ) }} \right ) ^{3}}{3\,k\ln \left ( a \right ) }}-{\frac{ \left ({{\rm e}^{lx\ln \left ( a \right ) }} \right ) ^{3}}{3\,l\ln \left ( a \right ) }}+3\,{\frac{{{\rm e}^{kx\ln \left ( a \right ) }} \left ({{\rm e}^{lx\ln \left ( a \right ) }} \right ) ^{2}}{\ln \left ( a \right ) \left ( k+2\,l \right ) }}-3\,{\frac{ \left ({{\rm e}^{kx\ln \left ( a \right ) }} \right ) ^{2}{{\rm e}^{lx\ln \left ( a \right ) }}}{\ln \left ( a \right ) \left ( 2\,k+l \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a^(k*x)-a^(l*x))^3,x)

[Out]

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

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Maxima [A]  time = 1.34094, size = 104, normalized size = 1.32 \[ -\frac{3 \, a^{2 \, k x + l x}}{{\left (2 \, k + l\right )} \log \left (a\right )} + \frac{3 \, a^{k x + 2 \, l x}}{{\left (k + 2 \, l\right )} \log \left (a\right )} + \frac{a^{3 \, k x}}{3 \, k \log \left (a\right )} - \frac{a^{3 \, l x}}{3 \, l \log \left (a\right )} \]

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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Fricas [A]  time = 0.218901, size = 177, normalized size = 2.24 \[ \frac{9 \,{\left (2 \, k^{2} l + k l^{2}\right )} a^{k x} a^{2 \, l x} - 9 \,{\left (k^{2} l + 2 \, k l^{2}\right )} a^{2 \, k x} a^{l x} +{\left (2 \, k^{2} l + 5 \, k l^{2} + 2 \, l^{3}\right )} a^{3 \, k x} -{\left (2 \, k^{3} + 5 \, k^{2} l + 2 \, k l^{2}\right )} a^{3 \, l x}}{3 \,{\left (2 \, k^{3} l + 5 \, k^{2} l^{2} + 2 \, k l^{3}\right )} \log \left (a\right )} \]

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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Sympy [A]  time = 56.9311, size = 663, normalized size = 8.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a**(k*x)-a**(l*x))**3,x)

[Out]

Piecewise((0, 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*log(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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GIAC/XCAS [A]  time = 0.235738, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^(k*x) - a^(l*x))^3,x, algorithm="giac")

[Out]

Done