Integrand size = 23, antiderivative size = 34 \[ \int \frac {f^{a+2 b x}}{c+d f^{e+2 b x}} \, dx=\frac {f^{a-e} \log \left (c+d f^{e+2 b x}\right )}{2 b d \log (f)} \]
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Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2279, 2278, 31} \[ \int \frac {f^{a+2 b x}}{c+d f^{e+2 b x}} \, dx=\frac {f^{a-e} \log \left (d f^{2 b x+e}+c\right )}{2 b d \log (f)} \]
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Rule 31
Rule 2278
Rule 2279
Rubi steps \begin{align*} \text {integral}& = f^{a-e} \int \frac {f^{e+2 b x}}{c+d f^{e+2 b x}} \, dx \\ & = \frac {f^{a-e} \text {Subst}\left (\int \frac {1}{c+d x} \, dx,x,f^{e+2 b x}\right )}{2 b \log (f)} \\ & = \frac {f^{a-e} \log \left (c+d f^{e+2 b x}\right )}{2 b d \log (f)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {f^{a+2 b x}}{c+d f^{e+2 b x}} \, dx=\frac {f^{a-e} \log \left (c+d f^{e+2 b x}\right )}{2 b d \log (f)} \]
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Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38
method | result | size |
norman | \(\frac {f^{-e} f^{a} \ln \left (c +d \,{\mathrm e}^{-\ln \left (f \right ) a +\ln \left (f \right ) e} {\mathrm e}^{\left (2 b x +a \right ) \ln \left (f \right )}\right )}{2 d \ln \left (f \right ) b}\) | \(47\) |
risch | \(-\frac {f^{a} f^{-e} a}{2 b d}+\frac {f^{a} f^{-e} \ln \left (f^{2 b x +a}+\frac {c \,f^{a} f^{-e}}{d}\right )}{2 \ln \left (f \right ) b d}\) | \(62\) |
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Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {f^{a+2 b x}}{c+d f^{e+2 b x}} \, dx=\frac {f^{a - e} \log \left (d f^{2 \, b x + e} + c\right )}{2 \, b d \log \left (f\right )} \]
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Time = 0.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int \frac {f^{a+2 b x}}{c+d f^{e+2 b x}} \, dx=\frac {e^{\left (a - e\right ) \log {\left (f \right )}} \log {\left (\frac {c e^{a \log {\left (f \right )}} e^{- e \log {\left (f \right )}}}{d} + f^{a + 2 b x} \right )}}{2 b d \log {\left (f \right )}} \]
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Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {f^{a+2 b x}}{c+d f^{e+2 b x}} \, dx=\frac {f^{a - e} \log \left (d f^{2 \, b x + e} + c\right )}{2 \, b d \log \left (f\right )} \]
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Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {f^{a+2 b x}}{c+d f^{e+2 b x}} \, dx=\frac {f^{a} \log \left ({\left | d f^{2 \, b x} f^{e} + c \right |}\right )}{2 \, b d f^{e} \log \left (f\right )} \]
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Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {f^{a+2 b x}}{c+d f^{e+2 b x}} \, dx=\frac {f^{a-e}\,\ln \left (d\,f^{a+e+2\,b\,x}+c\,f^a\right )}{2\,b\,d\,\ln \left (f\right )} \]
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