Integrand size = 21, antiderivative size = 27 \[ \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx=\frac {\left (a+b e^{c+d x}\right )^{1+n}}{b d (1+n)} \]
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Time = 0.02 (sec) , antiderivative size = 27, 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} \[ \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx=\frac {\left (a+b e^{c+d x}\right )^{n+1}}{b d (n+1)} \]
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Rule 32
Rule 2278
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+b x)^n \, dx,x,e^{c+d x}\right )}{d} \\ & = \frac {\left (a+b e^{c+d x}\right )^{1+n}}{b d (1+n)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx=\frac {\left (a+b e^{c+d x}\right )^{1+n}}{b d+b d n} \]
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Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\left (a +b \,{\mathrm e}^{d x +c}\right )^{1+n}}{b d \left (1+n \right )}\) | \(27\) |
default | \(\frac {\left (a +b \,{\mathrm e}^{d x +c}\right )^{1+n}}{b d \left (1+n \right )}\) | \(27\) |
risch | \(\frac {\left (a +b \,{\mathrm e}^{d x +c}\right ) \left (a +b \,{\mathrm e}^{d x +c}\right )^{n}}{b d \left (1+n \right )}\) | \(35\) |
parallelrisch | \(\frac {{\mathrm e}^{d x +c} \left (a +b \,{\mathrm e}^{d x +c}\right )^{n} b +\left (a +b \,{\mathrm e}^{d x +c}\right )^{n} a}{b d \left (1+n \right )}\) | \(48\) |
norman | \(\frac {{\mathrm e}^{d x +c} {\mathrm e}^{n \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}}{d \left (1+n \right )}+\frac {a \,{\mathrm e}^{n \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}}{b d \left (1+n \right )}\) | \(58\) |
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx=\frac {{\left (b e^{\left (d x + c\right )} + a\right )} {\left (b e^{\left (d x + c\right )} + a\right )}^{n}}{b d n + b d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (19) = 38\).
Time = 6.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.96 \[ \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx=\begin {cases} \frac {x e^{c}}{a} & \text {for}\: b = 0 \wedge d = 0 \wedge n = -1 \\\frac {a^{n} e^{c} e^{d x}}{d} & \text {for}\: b = 0 \\x \left (a + b e^{c}\right )^{n} e^{c} & \text {for}\: d = 0 \\\frac {\log {\left (\frac {a e^{- c}}{b} + e^{d x} \right )}}{b d} & \text {for}\: n = -1 \\\frac {a \left (a + b e^{c} e^{d x}\right )^{n}}{b d n + b d} + \frac {b \left (a + b e^{c} e^{d x}\right )^{n} e^{c} e^{d x}}{b d n + b d} & \text {otherwise} \end {cases} \]
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx=\frac {{\left (b e^{\left (d x + c\right )} + a\right )}^{n + 1}}{b d {\left (n + 1\right )}} \]
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Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx=\frac {{\left (b e^{\left (d x + c\right )} + a\right )}^{n + 1}}{b d {\left (n + 1\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int e^{c+d x} \left (a+b e^{c+d x}\right )^n \, dx=\frac {{\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}^{n+1}}{b\,d\,\left (n+1\right )} \]
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