Optimal. Leaf size=34 \[ \frac {f^{a-e} \log \left (d f^{2 b x+e}+c\right )}{2 b d \log (f)} \]
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Rubi [A] time = 0.08, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2247, 2246, 31} \[ \frac {f^{a-e} \log \left (d f^{2 b x+e}+c\right )}{2 b d \log (f)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2246
Rule 2247
Rubi steps
\begin {align*} \int \frac {f^{a+2 b x}}{c+d f^{e+2 b x}} \, dx &=f^{a-e} \int \frac {f^{e+2 b x}}{c+d f^{e+2 b x}} \, dx\\ &=\frac {f^{a-e} \operatorname {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*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 1.00 \[ \frac {f^{a-e} \log \left (d f^{2 b x+e}+c\right )}{2 b d \log (f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 32, normalized size = 0.94 \[ \frac {f^{a - e} \log \left (d f^{2 \, b x + e} + c\right )}{2 \, b d \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 37, normalized size = 1.09 \[ \frac {f^{a} \log \left ({\left | d f^{2 \, b x} f^{e} + c \right |}\right )}{2 \, b d f^{e} \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 47, normalized size = 1.38 \[ \frac {f^{a} f^{-e} \ln \left (d \,{\mathrm e}^{\left (2 b x +a \right ) \ln \relax (f )} {\mathrm e}^{-a \ln \relax (f )+e \ln \relax (f )}+c \right )}{2 b d \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 32, normalized size = 0.94 \[ \frac {f^{a - e} \log \left (d f^{2 \, b x + e} + c\right )}{2 \, b d \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 37, normalized size = 1.09 \[ \frac {f^{a-e}\,\ln \left (d\,f^{a+e+2\,b\,x}+c\,f^a\right )}{2\,b\,d\,\ln \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.77, size = 42, normalized size = 1.24 \[ \frac {e^{\left (a - e\right ) \log {\relax (f )}} \log {\left (\frac {c e^{a \log {\relax (f )}} e^{- e \log {\relax (f )}}}{d} + f^{a + 2 b x} \right )}}{2 b d \log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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