Integrand size = 21, antiderivative size = 30 \[ \int e^{n x} \left (a+b e^{n x}\right )^{r/s} \, dx=\frac {\left (a+b e^{n x}\right )^{\frac {r+s}{s}} s}{b n (r+s)} \]
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Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2278, 32} \[ \int e^{n x} \left (a+b e^{n x}\right )^{r/s} \, dx=\frac {s \left (a+b e^{n x}\right )^{\frac {r+s}{s}}}{b n (r+s)} \]
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Rule 32
Rule 2278
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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*}
Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int e^{n x} \left (a+b e^{n x}\right )^{r/s} \, dx=\frac {\left (a+b e^{n x}\right )^{1+\frac {r}{s}} s}{b n r+b n s} \]
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Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 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 | \(\frac {s \left (a +b \,{\mathrm e}^{n x}\right ) \left (a +b \,{\mathrm e}^{n x}\right )^{\frac {r}{s}}}{b n \left (r +s \right )}\) | \(36\) |
parallelrisch | \(\frac {{\mathrm e}^{n x} \left (a +b \,{\mathrm e}^{n x}\right )^{\frac {r}{s}} b s +\left (a +b \,{\mathrm e}^{n x}\right )^{\frac {r}{s}} a s}{b n \left (r +s \right )}\) | \(52\) |
norman | \(\frac {s \,{\mathrm e}^{n x} {\mathrm e}^{\frac {r \ln \left (a +b \,{\mathrm e}^{n x}\right )}{s}}}{n \left (r +s \right )}+\frac {a s \,{\mathrm e}^{\frac {r \ln \left (a +b \,{\mathrm e}^{n x}\right )}{s}}}{b n \left (r +s \right )}\) | \(60\) |
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Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int e^{n x} \left (a+b e^{n x}\right )^{r/s} \, dx=\frac {{\left (b s e^{\left (n x\right )} + a s\right )} {\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s}}}{b n r + b n s} \]
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (20) = 40\).
Time = 0.42 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.13 \[ \int e^{n x} \left (a+b e^{n x}\right )^{r/s} \, dx=\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \wedge n = 0 \wedge r = - s \\\frac {a^{\frac {r}{s}} e^{n x}}{n} & \text {for}\: b = 0 \\x \left (a + b\right )^{\frac {r}{s}} & \text {for}\: n = 0 \\\frac {\log {\left (\frac {a}{b} + e^{n x} \right )}}{b n} & \text {for}\: r = - s \\\frac {a s \left (a + b e^{n x}\right )^{\frac {r}{s}}}{b n r + b n s} + \frac {b s \left (a + b e^{n x}\right )^{\frac {r}{s}} e^{n x}}{b n r + b n s} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int e^{n x} \left (a+b e^{n x}\right )^{r/s} \, dx=\frac {{\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s} + 1}}{b n {\left (\frac {r}{s} + 1\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int e^{n x} \left (a+b e^{n x}\right )^{r/s} \, dx=\frac {{\left (b e^{\left (n x\right )} + a\right )}^{\frac {r}{s} + 1}}{b n {\left (\frac {r}{s} + 1\right )}} \]
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Time = 0.46 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int e^{n x} \left (a+b e^{n x}\right )^{r/s} \, dx=\frac {s\,{\left (a+b\,{\mathrm {e}}^{n\,x}\right )}^{\frac {r}{s}+1}}{b\,n\,\left (r+s\right )} \]
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