\(\int (a^{k x}+a^{l x})^n \, dx\) [506]

Optimal result

Integrand size = 13, antiderivative size = 72 \[ \int \left (a^{k x}+a^{l x}\right )^n \, dx=\frac {\left (1+a^{(k-l) x}\right ) \left (a^{k x}+a^{l x}\right )^n \operatorname {Hypergeometric2F1}\left (1,1+\frac {k n}{k-l},1+\frac {l n}{k-l},-a^{(k-l) x}\right )}{l n \log (a)} \]

[Out]

(1+a^((k-l)*x))*(a^(k*x)+a^(l*x))^n*hypergeom([1, 1+k*n/(k-l)],[1+l*n/(k-l)],-a^((k-l)*x))/l/n/ln(a)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2323, 2283} \[ \int \left (a^{k x}+a^{l x}\right )^n \, dx=\frac {\left (a^{-(x (k-l))}+1\right )^{-n} \left (a^{k x}+a^{l x}\right )^n \operatorname {Hypergeometric2F1}\left (-n,-\frac {k n}{k-l},1-\frac {k n}{k-l},-a^{-((k-l) x)}\right )}{k n \log (a)} \]

[In]

Int[(a^(k*x) + a^(l*x))^n,x]

[Out]

((a^(k*x) + a^(l*x))^n*Hypergeometric2F1[-n, -((k*n)/(k - l)), 1 - (k*n)/(k - l), -a^(-((k - l)*x))])/((1 + a^
(-((k - l)*x)))^n*k*n*Log[a])

Rule 2283

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[a^p*(G^(h*(f + g*x))/(g*h*Log[G]))*Hypergeometric2F1[-p, g*h*(Log[G]/(d*e*Log[F])), g*h*(Log[G]/(d*e*Log[F]))
+ 1, Simplify[(-b/a)*F^(e*(c + d*x))]], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] || G
tQ[a, 0])

Rule 2323

Int[(u_.)*((a_.)*(F_)^(v_) + (b_.)*(F_)^(w_))^(n_), x_Symbol] :> Dist[(a*F^v + b*F^w)^n/(F^(n*v)*(a + b*F^Expa
ndToSum[w - v, x])^n), Int[u*F^(n*v)*(a + b*F^ExpandToSum[w - v, x])^n, x], x] /; FreeQ[{F, a, b, n}, x] &&  !
IntegerQ[n] && LinearQ[{v, w}, x]

Rubi steps \begin{align*} \text {integral}& = \left (a^{-k n x} \left (1+a^{-((k-l) x)}\right )^{-n} \left (a^{k x}+a^{l x}\right )^n\right ) \int a^{k n x} \left (1+a^{-((k-l) x)}\right )^n \, dx \\ & = \frac {\left (1+a^{-((k-l) x)}\right )^{-n} \left (a^{k x}+a^{l x}\right )^n \operatorname {Hypergeometric2F1}\left (-n,-\frac {k n}{k-l},1-\frac {k n}{k-l},-a^{-((k-l) x)}\right )}{k n \log (a)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01 \[ \int \left (a^{k x}+a^{l x}\right )^n \, dx=\frac {\left (a^{k x}+a^{l x}\right )^n \left (1+a^{(-k+l) x}\right ) \operatorname {Hypergeometric2F1}\left (1,1+n+\frac {k n}{-k+l},1+\frac {k n}{-k+l},-a^{(-k+l) x}\right )}{k n \log (a)} \]

[In]

Integrate[(a^(k*x) + a^(l*x))^n,x]

[Out]

((a^(k*x) + a^(l*x))^n*(1 + a^((-k + l)*x))*Hypergeometric2F1[1, 1 + n + (k*n)/(-k + l), 1 + (k*n)/(-k + l), -
a^((-k + l)*x)])/(k*n*Log[a])

Maple [F]

\[\int \left (a^{k x}+a^{l x}\right )^{n}d x\]

[In]

int((a^(k*x)+a^(l*x))^n,x)

[Out]

int((a^(k*x)+a^(l*x))^n,x)

Fricas [F]

\[ \int \left (a^{k x}+a^{l x}\right )^n \, dx=\int { {\left (a^{k x} + a^{l x}\right )}^{n} \,d x } \]

[In]

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

[Out]

integral((a^(k*x) + a^(l*x))^n, x)

Sympy [F]

\[ \int \left (a^{k x}+a^{l x}\right )^n \, dx=\int \left (a^{k x} + a^{l x}\right )^{n}\, dx \]

[In]

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

[Out]

Integral((a**(k*x) + a**(l*x))**n, x)

Maxima [F]

\[ \int \left (a^{k x}+a^{l x}\right )^n \, dx=\int { {\left (a^{k x} + a^{l x}\right )}^{n} \,d x } \]

[In]

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

[Out]

integrate((a^(k*x) + a^(l*x))^n, x)

Giac [F]

\[ \int \left (a^{k x}+a^{l x}\right )^n \, dx=\int { {\left (a^{k x} + a^{l x}\right )}^{n} \,d x } \]

[In]

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

[Out]

integrate((a^(k*x) + a^(l*x))^n, x)

Mupad [F(-1)]

Timed out. \[ \int \left (a^{k x}+a^{l x}\right )^n \, dx=\int {\left (a^{k\,x}+a^{l\,x}\right )}^n \,d x \]

[In]

int((a^(k*x) + a^(l*x))^n,x)

[Out]

int((a^(k*x) + a^(l*x))^n, x)