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

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

[Out]

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

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Rubi [A]  time = 0.065609, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{s \left (a+b e^{n x}\right )^{\frac{r+s}{s}}}{b n (r+s)} \]

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))

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Rubi in Sympy [A]  time = 5.2087, size = 20, normalized size = 0.67 \[ \frac{s \left (a + b e^{n x}\right )^{\frac{r + s}{s}}}{b n \left (r + s\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(n*x)*(a+b*exp(n*x))**(r/s),x)

[Out]

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

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Mathematica [A]  time = 0.0479255, size = 30, normalized size = 1. \[ \frac{s \left (a+b e^{n x}\right )^{\frac{r}{s}+1}}{b n r+b n s} \]

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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Maple [A]  time = 0.003, size = 33, normalized size = 1.1 \[{\frac{1}{nb} \left ( a+b{{\rm e}^{nx}} \right ) ^{{\frac{r}{s}}+1} \left ({\frac{r}{s}}+1 \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(n*x) + a)^(r/s)*e^(n*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22827, size = 50, normalized size = 1.67 \[ \frac{{\left (b s e^{\left (n x\right )} + a s\right )}{\left (b e^{\left (n x\right )} + a\right )}^{\frac{r}{s}}}{b n r + b n s} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(n*x) + a)^(r/s)*e^(n*x),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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Sympy [A]  time = 2.05022, 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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GIAC/XCAS [A]  time = 0.20331, size = 43, normalized size = 1.43 \[ \frac{{\left (b e^{\left (n x\right )} + a\right )}^{\frac{r}{s} + 1}}{b n{\left (\frac{r}{s} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(n*x) + a)^(r/s)*e^(n*x),x, algorithm="giac")

[Out]

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