Integrand size = 17, antiderivative size = 37 \[ \int e^{n x} \left (a+b \left (e^x\right )^n\right )^p \, dx=\frac {e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{1+p}}{b n (1+p)} \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2279, 2278, 32} \[ \int e^{n x} \left (a+b \left (e^x\right )^n\right )^p \, dx=\frac {e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]
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Rule 32
Rule 2278
Rule 2279
Rubi steps \begin{align*} \text {integral}& = \left (e^{n x} \left (e^x\right )^{-n}\right ) \int \left (e^x\right )^n \left (a+b \left (e^x\right )^n\right )^p \, dx \\ & = \frac {\left (e^{n x} \left (e^x\right )^{-n}\right ) \text {Subst}\left (\int (a+b x)^p \, dx,x,\left (e^x\right )^n\right )}{n} \\ & = \frac {e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{1+p}}{b n (1+p)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97 \[ \int e^{n x} \left (a+b \left (e^x\right )^n\right )^p \, dx=\frac {e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{1+p}}{b n+b n p} \]
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Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {\left (a +b \,{\mathrm e}^{n x}\right ) \left (a +b \,{\mathrm e}^{n x}\right )^{p}}{b n \left (p +1\right )}\) | \(31\) |
norman | \(\frac {{\mathrm e}^{n x} {\mathrm e}^{p \ln \left (a +b \,{\mathrm e}^{n x}\right )}}{n \left (p +1\right )}+\frac {a \,{\mathrm e}^{p \ln \left (a +b \,{\mathrm e}^{n x}\right )}}{b n \left (p +1\right )}\) | \(52\) |
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Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.78 \[ \int e^{n x} \left (a+b \left (e^x\right )^n\right )^p \, dx=\frac {{\left (b e^{\left (n x\right )} + a\right )} {\left (b e^{\left (n x\right )} + a\right )}^{p}}{b n p + b n} \]
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\[ \int e^{n x} \left (a+b \left (e^x\right )^n\right )^p \, dx=\int \left (a + b \left (e^{x}\right )^{n}\right )^{p} e^{n x}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65 \[ \int e^{n x} \left (a+b \left (e^x\right )^n\right )^p \, dx=\frac {{\left (b e^{\left (n x\right )} + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]
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Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65 \[ \int e^{n x} \left (a+b \left (e^x\right )^n\right )^p \, dx=\frac {{\left (b e^{\left (n x\right )} + a\right )}^{p + 1}}{b n {\left (p + 1\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65 \[ \int e^{n x} \left (a+b \left (e^x\right )^n\right )^p \, dx=\frac {{\left (a+b\,{\mathrm {e}}^{n\,x}\right )}^{p+1}}{b\,n\,\left (p+1\right )} \]
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