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

Optimal. Leaf size=30 \[ \frac {s \left (a+b e^{n x}\right )^{\frac {r+s}{s}}}{b n (r+s)} \]

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Rubi [A]  time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2246, 32} \[ \frac {s \left (a+b e^{n x}\right )^{\frac {r+s}{s}}}{b n (r+s)} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*x)*(a + b*E^(n*x))^(r/s),x]

[Out]

((a + b*E^(n*x))^((r + s)/s)*s)/(b*n*(r + s))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2246

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, x]

Rubi steps

\begin {align*} \int e^{n x} \left (a+b e^{n x}\right )^{r/s} \, dx &=\frac {\operatorname {Subst}\left (\int (a+b x)^{r/s} \, dx,x,e^{n x}\right )}{n}\\ &=\frac {\left (a+b e^{n x}\right )^{\frac {r+s}{s}} s}{b n (r+s)}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 30, normalized size = 1.00 \[ \frac {s \left (a+b e^{n x}\right )^{\frac {r}{s}+1}}{b n r+b n s} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*x)*(a + b*E^(n*x))^(r/s),x]

[Out]

((a + b*E^(n*x))^(1 + r/s)*s)/(b*n*r + b*n*s)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{n x} \left (a+b e^{n x}\right )^{r/s} \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[E^(n*x)*(a + b*E^(n*x))^(r/s),x]

[Out]

Could not integrate

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fricas [A]  time = 1.30, size = 37, normalized size = 1.23 \[ \frac {{\left (b s e^{\left (n x\right )} + a s\right )} {\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s}}}{b n r + b n s} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*s*e^(n*x) + a*s)*(b*e^(n*x) + a)^(r/s)/(b*n*r + b*n*s)

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giac [A]  time = 0.64, size = 32, normalized size = 1.07 \[ \frac {{\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s} + 1}}{b n {\left (\frac {r}{s} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*e^(n*x) + a)^(r/s + 1)/(b*n*(r/s + 1))

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maple [A]  time = 0.04, size = 33, normalized size = 1.10

method	result	size
derivativedivides	\(\frac {\left (a +b \,{\mathrm e}^{n x}\right )^{\frac {r}{s}+1}}{n b \left (\frac {r}{s}+1\right )}\)	\(33\)
default	\(\frac {\left (a +b \,{\mathrm e}^{n x}\right )^{\frac {r}{s}+1}}{n b \left (\frac {r}{s}+1\right )}\)	\(33\)
risch	\(\frac {s \left (a +b \,{\mathrm e}^{n x}\right ) \left (a +b \,{\mathrm e}^{n x}\right )^{\frac {r}{s}}}{b n \left (r +s \right )}\)	\(36\)
norman	\(\frac {s \,{\mathrm e}^{n x} {\mathrm e}^{\frac {r \ln \left (a +b \,{\mathrm e}^{n x}\right )}{s}}}{n \left (r +s \right )}+\frac {a s \,{\mathrm e}^{\frac {r \ln \left (a +b \,{\mathrm e}^{n x}\right )}{s}}}{b n \left (r +s \right )}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/n*(a+b*exp(n*x))^(r/s+1)/b/(r/s+1)

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maxima [A]  time = 0.59, size = 32, normalized size = 1.07 \[ \frac {{\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s} + 1}}{b n {\left (\frac {r}{s} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*e^(n*x) + a)^(r/s + 1)/(b*n*(r/s + 1))

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mupad [B]  time = 0.35, size = 29, normalized size = 0.97 \[ \frac {s\,{\left (a+b\,{\mathrm {e}}^{n\,x}\right )}^{\frac {r}{s}+1}}{b\,n\,\left (r+s\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(s*(a + b*exp(n*x))^(r/s + 1))/(b*n*(r + s))

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sympy [A]  time = 1.32, size = 94, normalized size = 3.13 \[ \begin {cases} \frac {x}{a} & \text {for}\: b = 0 \wedge n = 0 \wedge r = - s \\\frac {a^{\frac {r}{s}} e^{n x}}{n} & \text {for}\: b = 0 \\x \left (a + b\right )^{\frac {r}{s}} & \text {for}\: n = 0 \\\frac {\log {\left (\frac {a}{b} + e^{n x} \right )}}{b n} & \text {for}\: r = - s \\\frac {a s \left (a + b e^{n x}\right )^{\frac {r}{s}}}{b n r + b n s} + \frac {b s \left (a + b e^{n x}\right )^{\frac {r}{s}} e^{n x}}{b n r + b n s} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/a, Eq(b, 0) & Eq(n, 0) & Eq(r, -s)), (a**(r/s)*exp(n*x)/n, Eq(b, 0)), (x*(a + b)**(r/s), Eq(n, 0)
), (log(a/b + exp(n*x))/(b*n), Eq(r, -s)), (a*s*(a + b*exp(n*x))**(r/s)/(b*n*r + b*n*s) + b*s*(a + b*exp(n*x))
**(r/s)*exp(n*x)/(b*n*r + b*n*s), True))

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