\(\int (a+b e^{n x})^{r/s} \, dx\) [529]

Optimal result

Integrand size = 15, antiderivative size = 59 \[ \int \left (a+b e^{n x}\right )^{r/s} \, dx=-\frac {\left (a+b e^{n x}\right )^{\frac {r+s}{s}} s \operatorname {Hypergeometric2F1}\left (1,\frac {r+s}{s},2+\frac {r}{s},1+\frac {b e^{n x}}{a}\right )}{a n (r+s)} \]

[Out]

-(a+b*exp(n*x))^((r+s)/s)*s*hypergeom([1, (r+s)/s],[2+r/s],1+b*exp(n*x)/a)/a/n/(r+s)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2320, 67} \[ \int \left (a+b e^{n x}\right )^{r/s} \, dx=-\frac {s \left (a+b e^{n x}\right )^{\frac {r+s}{s}} \operatorname {Hypergeometric2F1}\left (1,\frac {r+s}{s},\frac {r}{s}+2,\frac {e^{n x} b}{a}+1\right )}{a n (r+s)} \]

[In]

Int[(a + b*E^(n*x))^(r/s),x]

[Out]

-(((a + b*E^(n*x))^((r + s)/s)*s*Hypergeometric2F1[1, (r + s)/s, 2 + r/s, 1 + (b*E^(n*x))/a])/(a*n*(r + s)))

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b x)^{r/s}}{x} \, dx,x,e^{n x}\right )}{n} \\ & = -\frac {\left (a+b e^{n x}\right )^{\frac {r+s}{s}} s \operatorname {Hypergeometric2F1}\left (1,\frac {r+s}{s},2+\frac {r}{s},1+\frac {b e^{n x}}{a}\right )}{a n (r+s)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \left (a+b e^{n x}\right )^{r/s} \, dx=-\frac {\left (a+b e^{n x}\right )^{\frac {r+s}{s}} s \operatorname {Hypergeometric2F1}\left (1,\frac {r+s}{s},2+\frac {r}{s},1+\frac {b e^{n x}}{a}\right )}{a n (r+s)} \]

[In]

Integrate[(a + b*E^(n*x))^(r/s),x]

[Out]

-(((a + b*E^(n*x))^((r + s)/s)*s*Hypergeometric2F1[1, (r + s)/s, 2 + r/s, 1 + (b*E^(n*x))/a])/(a*n*(r + s)))

Maple [F]

\[\int \left (a +b \,{\mathrm e}^{n x}\right )^{\frac {r}{s}}d x\]

[In]

int((a+b*exp(n*x))^(r/s),x)

[Out]

int((a+b*exp(n*x))^(r/s),x)

Fricas [F]

\[ \int \left (a+b e^{n x}\right )^{r/s} \, dx=\int { {\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s}} \,d x } \]

[In]

integrate((a+b*exp(n*x))^(r/s),x, algorithm="fricas")

[Out]

integral((b*e^(n*x) + a)^(r/s), x)

Sympy [F]

\[ \int \left (a+b e^{n x}\right )^{r/s} \, dx=\int \left (a + b e^{n x}\right )^{\frac {r}{s}}\, dx \]

[In]

integrate((a+b*exp(n*x))**(r/s),x)

[Out]

Integral((a + b*exp(n*x))**(r/s), x)

Maxima [F]

\[ \int \left (a+b e^{n x}\right )^{r/s} \, dx=\int { {\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s}} \,d x } \]

[In]

integrate((a+b*exp(n*x))^(r/s),x, algorithm="maxima")

[Out]

integrate((b*e^(n*x) + a)^(r/s), x)

Giac [F]

\[ \int \left (a+b e^{n x}\right )^{r/s} \, dx=\int { {\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s}} \,d x } \]

[In]

integrate((a+b*exp(n*x))^(r/s),x, algorithm="giac")

[Out]

integrate((b*e^(n*x) + a)^(r/s), x)

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.27 \[ \int \left (a+b e^{n x}\right )^{r/s} \, dx=\frac {s\,{\left (a+b\,{\mathrm {e}}^{n\,x}\right )}^{r/s}\,{{}}_2{\mathrm {F}}_1\left (-\frac {r}{s},-\frac {r}{s};\ 1-\frac {r}{s};\ -\frac {a\,{\mathrm {e}}^{-n\,x}}{b}\right )}{n\,r\,{\left (\frac {a\,{\mathrm {e}}^{-n\,x}}{b}+1\right )}^{r/s}} \]

[In]

int((a + b*exp(n*x))^(r/s),x)

[Out]

(s*(a + b*exp(n*x))^(r/s)*hypergeom([-r/s, -r/s], 1 - r/s, -(a*exp(-n*x))/b))/(n*r*((a*exp(-n*x))/b + 1)^(r/s)
)