Integrand size = 13, antiderivative size = 79 \[ \int \left (a^{k x}+a^{l x}\right )^3 \, dx=\frac {a^{3 k x}}{3 k \log (a)}+\frac {a^{3 l x}}{3 l \log (a)}+\frac {3 a^{(2 k+l) x}}{(2 k+l) \log (a)}+\frac {3 a^{(k+2 l) x}}{(k+2 l) \log (a)} \] Output:
1/3*a^(3*k*x)/k/ln(a)+1/3*a^(3*l*x)/l/ln(a)+3*a^((2*k+l)*x)/(2*k+l)/ln(a)+ 3*a^((k+2*l)*x)/(k+2*l)/ln(a)
Time = 0.07 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.82 \[ \int \left (a^{k x}+a^{l x}\right )^3 \, dx=\frac {\frac {a^{3 k x}}{k}+\frac {a^{3 l x}}{l}+\frac {9 a^{(2 k+l) x}}{2 k+l}+\frac {9 a^{(k+2 l) x}}{k+2 l}}{3 \log (a)} \] Input:
Integrate[(a^(k*x) + a^(l*x))^3,x]
Output:
(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])
Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {7281, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a^{k x}+a^{l x}\right )^3 \, dx\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle \frac {\int \left (a^{k x}+a^{l x}\right )^3d(x \log (a))}{\log (a)}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {\int \left (a^{3 k x}+a^{3 l x}+3 a^{(2 k+l) x}+3 a^{(k+2 l) x}\right )d(x \log (a))}{\log (a)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {3 a^{x (2 k+l)}}{2 k+l}+\frac {3 a^{x (k+2 l)}}{k+2 l}+\frac {a^{3 k x}}{3 k}+\frac {a^{3 l x}}{3 l}}{\log (a)}\) |
Input:
Int[(a^(k*x) + a^(l*x))^3,x]
Output:
(a^(3*k*x)/(3*k) + a^(3*l*x)/(3*l) + (3*a^((2*k + l)*x))/(2*k + l) + (3*a^ ((k + 2*l)*x))/(k + 2*l))/Log[a]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.06
method | result | size |
risch | \(\frac {a^{3 k x}}{3 k \ln \left (a \right )}+\frac {a^{3 l x}}{3 l \ln \left (a \right )}+\frac {3 a^{k x} a^{2 l x}}{\ln \left (a \right ) \left (k +2 l \right )}+\frac {3 a^{2 k x} a^{l x}}{\ln \left (a \right ) \left (2 k +l \right )}\) | \(84\) |
norman | \(\frac {{\mathrm e}^{3 k x \ln \left (a \right )}}{3 k \ln \left (a \right )}+\frac {{\mathrm e}^{3 l x \ln \left (a \right )}}{3 l \ln \left (a \right )}+\frac {3 \,{\mathrm e}^{k x \ln \left (a \right )} {\mathrm e}^{2 l x \ln \left (a \right )}}{\ln \left (a \right ) \left (k +2 l \right )}+\frac {3 \,{\mathrm e}^{2 k x \ln \left (a \right )} {\mathrm e}^{l x \ln \left (a \right )}}{\ln \left (a \right ) \left (2 k +l \right )}\) | \(90\) |
meijerg | \(-\frac {1-{\mathrm e}^{3 k x \ln \left (a \right )}}{3 k \ln \left (a \right )}-\frac {3 \left (1-{\mathrm e}^{x k \ln \left (a \right ) \left (2+\frac {l}{k}\right )}\right )}{k \ln \left (a \right ) \left (2+\frac {l}{k}\right )}-\frac {3 \left (1-{\mathrm e}^{x l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {1}{1+\frac {k}{l}}\right )}\right )}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {1}{1+\frac {k}{l}}\right )}-\frac {1-{\mathrm e}^{3 l x \ln \left (a \right )}}{3 l \ln \left (a \right )}\) | \(136\) |
parallelrisch | \(\frac {2 a^{3 k x} k^{2} l +5 a^{3 k x} k \,l^{2}+2 a^{3 k x} l^{3}+9 a^{2 k x} a^{l x} k^{2} l +18 a^{2 k x} a^{l x} k \,l^{2}+18 a^{k x} a^{2 l x} k^{2} l +9 a^{k x} a^{2 l x} k \,l^{2}+2 a^{3 l x} k^{3}+5 a^{3 l x} k^{2} l +2 a^{3 l x} k \,l^{2}}{3 \ln \left (a \right ) k l \left (k +2 l \right ) \left (2 k +l \right )}\) | \(176\) |
orering | \(\frac {2 \left (k^{3}+8 l \,k^{2}+8 l^{2} k +l^{3}\right ) \left (a^{k x}+a^{l x}\right )^{3}}{3 \ln \left (a \right ) l \left (2 k^{2}+5 l k +2 l^{2}\right ) k}-\frac {\left (11 k^{2}+32 l k +11 l^{2}\right ) \left (a^{k x}+a^{l x}\right )^{2} \left (a^{k x} k \ln \left (a \right )+a^{l x} l \ln \left (a \right )\right )}{3 \ln \left (a \right )^{2} l \left (2 k^{2}+5 l k +2 l^{2}\right ) k}+\frac {2 \left (k +l \right ) \left (6 \left (a^{k x}+a^{l x}\right ) \left (a^{k x} k \ln \left (a \right )+a^{l x} l \ln \left (a \right )\right )^{2}+3 \left (a^{k x}+a^{l x}\right )^{2} \left (a^{k x} k^{2} \ln \left (a \right )^{2}+a^{l x} l^{2} \ln \left (a \right )^{2}\right )\right )}{3 \ln \left (a \right )^{3} l \left (2 k^{2}+5 l k +2 l^{2}\right ) k}-\frac {6 \left (a^{k x} k \ln \left (a \right )+a^{l x} l \ln \left (a \right )\right )^{3}+18 \left (a^{k x}+a^{l x}\right ) \left (a^{k x} k \ln \left (a \right )+a^{l x} l \ln \left (a \right )\right ) \left (a^{k x} k^{2} \ln \left (a \right )^{2}+a^{l x} l^{2} \ln \left (a \right )^{2}\right )+3 \left (a^{k x}+a^{l x}\right )^{2} \left (a^{k x} k^{3} \ln \left (a \right )^{3}+a^{l x} l^{3} \ln \left (a \right )^{3}\right )}{9 \ln \left (a \right )^{4} l \left (2 k^{2}+5 l k +2 l^{2}\right ) k}\) | \(402\) |
Input:
int((a^(k*x)+a^(l*x))^3,x,method=_RETURNVERBOSE)
Output:
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)
Time = 0.06 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.65 \[ \int \left (a^{k x}+a^{l x}\right )^3 \, dx=\frac {9 \, {\left (2 \, k^{2} l + k l^{2}\right )} a^{k x} a^{2 \, l x} + 9 \, {\left (k^{2} l + 2 \, k l^{2}\right )} a^{2 \, k x} a^{l x} + {\left (2 \, k^{2} l + 5 \, k l^{2} + 2 \, l^{3}\right )} a^{3 \, k x} + {\left (2 \, k^{3} + 5 \, k^{2} l + 2 \, k l^{2}\right )} a^{3 \, l x}}{3 \, {\left (2 \, k^{3} l + 5 \, k^{2} l^{2} + 2 \, k l^{3}\right )} \log \left (a\right )} \] Input:
integrate((a^(k*x)+a^(l*x))^3,x, algorithm="fricas")
Output:
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))
Leaf count of result is larger than twice the leaf count of optimal. 665 vs. \(2 (63) = 126\).
Time = 1.82 (sec) , antiderivative size = 665, normalized size of antiderivative = 8.42 \[ \int \left (a^{k x}+a^{l x}\right )^3 \, dx =\text {Too large to display} \] Input:
integrate((a**(k*x)+a**(l*x))**3,x)
Output:
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 - 1/(a**(3*l*x)*l*log(a )) - 1/(6*a**(6*l*x)*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/(3*a**(3*l*x/2)*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*lo g(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))
Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97 \[ \int \left (a^{k x}+a^{l x}\right )^3 \, dx=\frac {3 \, a^{2 \, k x + l x}}{{\left (2 \, k + l\right )} \log \left (a\right )} + \frac {3 \, a^{k x + 2 \, l x}}{{\left (k + 2 \, l\right )} \log \left (a\right )} + \frac {a^{3 \, k x}}{3 \, k \log \left (a\right )} + \frac {a^{3 \, l x}}{3 \, l \log \left (a\right )} \] Input:
integrate((a^(k*x)+a^(l*x))^3,x, algorithm="maxima")
Output:
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))
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 1033, normalized size of antiderivative = 13.08 \[ \int \left (a^{k x}+a^{l x}\right )^3 \, dx=\text {Too large to display} \] Input:
integrate((a^(k*x)+a^(l*x))^3,x, algorithm="giac")
Output:
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*l og(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))*a bs(a)^(3*l*x) + I*abs(a)^(3*k*x)*(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(abs(a))) - 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(abs(a)))) + I*abs (a)^(3*l*x)*(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))) - 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) + 3*I*(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))) - I*e^(-I*pi*k*x...
Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.03 \[ \int \left (a^{k x}+a^{l x}\right )^3 \, dx=\frac {3\,a^{k\,x}\,a^{2\,l\,x}}{k\,\ln \left (a\right )+2\,l\,\ln \left (a\right )}+\frac {3\,a^{2\,k\,x}\,a^{l\,x}}{2\,k\,\ln \left (a\right )+l\,\ln \left (a\right )}+\frac {a^{3\,k\,x}}{3\,k\,\ln \left (a\right )}+\frac {a^{3\,l\,x}}{3\,l\,\ln \left (a\right )} \] Input:
int((a^(k*x) + a^(l*x))^3,x)
Output:
(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))
Time = 0.16 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.08 \[ \int \left (a^{k x}+a^{l x}\right )^3 \, dx=\frac {2 a^{3 k x} k^{2} l +5 a^{3 k x} k \,l^{2}+2 a^{3 k x} l^{3}+9 a^{2 k x +l x} k^{2} l +18 a^{2 k x +l x} k \,l^{2}+18 a^{k x +2 l x} k^{2} l +9 a^{k x +2 l x} k \,l^{2}+2 a^{3 l x} k^{3}+5 a^{3 l x} k^{2} l +2 a^{3 l x} k \,l^{2}}{3 \,\mathrm {log}\left (a \right ) k l \left (2 k^{2}+5 k l +2 l^{2}\right )} \] Input:
int((a^(k*x)+a^(l*x))^3,x)
Output:
(2*a**(3*k*x)*k**2*l + 5*a**(3*k*x)*k*l**2 + 2*a**(3*k*x)*l**3 + 9*a**(2*k *x + l*x)*k**2*l + 18*a**(2*k*x + l*x)*k*l**2 + 18*a**(k*x + 2*l*x)*k**2*l + 9*a**(k*x + 2*l*x)*k*l**2 + 2*a**(3*l*x)*k**3 + 5*a**(3*l*x)*k**2*l + 2 *a**(3*l*x)*k*l**2)/(3*log(a)*k*l*(2*k**2 + 5*k*l + 2*l**2))