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

Optimal. Leaf size=59 \[ -\frac{s \left (a+b e^{n x}\right )^{\frac{r+s}{s}} \, _2F_1\left (1,\frac{r+s}{s};\frac{r}{s}+2;\frac{e^{n x} b}{a}+1\right )}{a n (r+s)} \]

[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)))

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Rubi [A]  time = 0.0305085, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2282, 65} \[ -\frac{s \left (a+b e^{n x}\right )^{\frac{r+s}{s}} \text{Hypergeometric2F1}\left (1,\frac{r+s}{s},\frac{r}{s}+2,\frac{b e^{n x}}{a}+1\right )}{a n (r+s)} \]

Antiderivative was successfully verified.

[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 2282

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]]]

Rule 65

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

Rubi steps

\begin{align*} \int \left (a+b e^{n x}\right )^{r/s} \, dx &=\frac{\operatorname{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 \, _2F_1\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]  time = 0.0213791, size = 59, normalized size = 1. \[ -\frac{s \left (a+b e^{n x}\right )^{\frac{r+s}{s}} \, _2F_1\left (1,\frac{r+s}{s};\frac{r}{s}+2;\frac{e^{n x} b}{a}+1\right )}{a n (r+s)} \]

Antiderivative was successfully verified.

[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)))

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Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{{\rm e}^{nx}} \right ) ^{{\frac{r}{s}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b e^{\left (n x\right )} + a\right )}^{\frac{r}{s}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*e^(n*x) + a)^(r/s), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b e^{\left (n x\right )} + a\right )}^{\frac{r}{s}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*e^(n*x) + a)^(r/s), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b e^{n x}\right )^{\frac{r}{s}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*exp(n*x))**(r/s), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b e^{\left (n x\right )} + a\right )}^{\frac{r}{s}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*e^(n*x) + a)^(r/s), x)