3.510 \(\int \left (a^{k x}-a^{l x}\right )^4 \, dx\)

Optimal. Leaf size=98 \[ \frac{3 a^{2 x (k+l)}}{\log (a) (k+l)}-\frac{4 a^{x (3 k+l)}}{\log (a) (3 k+l)}-\frac{4 a^{x (k+3 l)}}{\log (a) (k+3 l)}+\frac{a^{4 k x}}{4 k \log (a)}+\frac{a^{4 l x}}{4 l \log (a)} \]

[Out]

a^(4*k*x)/(4*k*Log[a]) + a^(4*l*x)/(4*l*Log[a]) + (3*a^(2*(k + l)*x))/((k + l)*L
og[a]) - (4*a^((3*k + l)*x))/((3*k + l)*Log[a]) - (4*a^((k + 3*l)*x))/((k + 3*l)
*Log[a])

_______________________________________________________________________________________

Rubi [A]  time = 0.207649, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{3 a^{2 x (k+l)}}{\log (a) (k+l)}-\frac{4 a^{x (3 k+l)}}{\log (a) (3 k+l)}-\frac{4 a^{x (k+3 l)}}{\log (a) (k+3 l)}+\frac{a^{4 k x}}{4 k \log (a)}+\frac{a^{4 l x}}{4 l \log (a)} \]

Antiderivative was successfully verified.

[In]  Int[(a^(k*x) - a^(l*x))^4,x]

[Out]

a^(4*k*x)/(4*k*Log[a]) + a^(4*l*x)/(4*l*Log[a]) + (3*a^(2*(k + l)*x))/((k + l)*L
og[a]) - (4*a^((3*k + l)*x))/((3*k + l)*Log[a]) - (4*a^((k + 3*l)*x))/((k + 3*l)
*Log[a])

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a^{k x} - a^{l x}\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a**(k*x)-a**(l*x))**4,x)

[Out]

Integral((a**(k*x) - a**(l*x))**4, x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.110412, size = 80, normalized size = 0.82 \[ \frac{\frac{12 a^{2 x (k+l)}}{k+l}-\frac{16 a^{x (3 k+l)}}{3 k+l}-\frac{16 a^{x (k+3 l)}}{k+3 l}+\frac{a^{4 k x}}{k}+\frac{a^{4 l x}}{l}}{4 \log (a)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^(k*x) - a^(l*x))^4,x]

[Out]

(a^(4*k*x)/k + a^(4*l*x)/l + (12*a^(2*(k + l)*x))/(k + l) - (16*a^((3*k + l)*x))
/(3*k + l) - (16*a^((k + 3*l)*x))/(k + 3*l))/(4*Log[a])

_______________________________________________________________________________________

Maple [A]  time = 0.026, size = 109, normalized size = 1.1 \[{\frac{ \left ({a}^{kx} \right ) ^{4}}{4\,k\ln \left ( a \right ) }}-4\,{\frac{ \left ({a}^{kx} \right ) ^{3}{a}^{lx}}{\ln \left ( a \right ) \left ( 3\,k+l \right ) }}+3\,{\frac{ \left ({a}^{kx} \right ) ^{2} \left ({a}^{lx} \right ) ^{2}}{\ln \left ( a \right ) \left ( k+l \right ) }}-4\,{\frac{{a}^{kx} \left ({a}^{lx} \right ) ^{3}}{\ln \left ( a \right ) \left ( k+3\,l \right ) }}+{\frac{ \left ({a}^{lx} \right ) ^{4}}{4\,l\ln \left ( a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a^(k*x)-a^(l*x))^4,x)

[Out]

1/4/ln(a)/k*(a^(k*x))^4-4*(a^(k*x))^3/ln(a)/(3*k+l)*a^(l*x)+3*(a^(k*x))^2/ln(a)/
(k+l)*(a^(l*x))^2-4*a^(k*x)/ln(a)/(k+3*l)*(a^(l*x))^3+1/4/ln(a)/l*(a^(l*x))^4

_______________________________________________________________________________________

Maxima [A]  time = 1.34728, size = 134, normalized size = 1.37 \[ -\frac{4 \, a^{3 \, k x + l x}}{{\left (3 \, k + l\right )} \log \left (a\right )} - \frac{4 \, a^{k x + 3 \, l x}}{{\left (k + 3 \, l\right )} \log \left (a\right )} + \frac{3 \, a^{2 \, k x + 2 \, l x}}{{\left (k + l\right )} \log \left (a\right )} + \frac{a^{4 \, k x}}{4 \, k \log \left (a\right )} + \frac{a^{4 \, l x}}{4 \, l \log \left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^(k*x) - a^(l*x))^4,x, algorithm="maxima")

[Out]

-4*a^(3*k*x + l*x)/((3*k + l)*log(a)) - 4*a^(k*x + 3*l*x)/((k + 3*l)*log(a)) + 3
*a^(2*k*x + 2*l*x)/((k + l)*log(a)) + 1/4*a^(4*k*x)/(k*log(a)) + 1/4*a^(4*l*x)/(
l*log(a))

_______________________________________________________________________________________

Fricas [A]  time = 0.215607, size = 279, normalized size = 2.85 \[ -\frac{16 \,{\left (3 \, k^{3} l + 4 \, k^{2} l^{2} + k l^{3}\right )} a^{k x} a^{3 \, l x} - 12 \,{\left (3 \, k^{3} l + 10 \, k^{2} l^{2} + 3 \, k l^{3}\right )} a^{2 \, k x} a^{2 \, l x} + 16 \,{\left (k^{3} l + 4 \, k^{2} l^{2} + 3 \, k l^{3}\right )} a^{3 \, k x} a^{l x} -{\left (3 \, k^{3} l + 13 \, k^{2} l^{2} + 13 \, k l^{3} + 3 \, l^{4}\right )} a^{4 \, k x} -{\left (3 \, k^{4} + 13 \, k^{3} l + 13 \, k^{2} l^{2} + 3 \, k l^{3}\right )} a^{4 \, l x}}{4 \,{\left (3 \, k^{4} l + 13 \, k^{3} l^{2} + 13 \, k^{2} l^{3} + 3 \, k l^{4}\right )} \log \left (a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^(k*x) - a^(l*x))^4,x, algorithm="fricas")

[Out]

-1/4*(16*(3*k^3*l + 4*k^2*l^2 + k*l^3)*a^(k*x)*a^(3*l*x) - 12*(3*k^3*l + 10*k^2*
l^2 + 3*k*l^3)*a^(2*k*x)*a^(2*l*x) + 16*(k^3*l + 4*k^2*l^2 + 3*k*l^3)*a^(3*k*x)*
a^(l*x) - (3*k^3*l + 13*k^2*l^2 + 13*k*l^3 + 3*l^4)*a^(4*k*x) - (3*k^4 + 13*k^3*
l + 13*k^2*l^2 + 3*k*l^3)*a^(4*l*x))/((3*k^4*l + 13*k^3*l^2 + 13*k^2*l^3 + 3*k*l
^4)*log(a))

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a**(k*x)-a**(l*x))**4,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.266916, 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))^4,x, algorithm="giac")

[Out]

Done