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 = 0.0811105, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, 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 2247
Rule 2246
Rule 31
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*}
Mathematica [A] time = 0.017154, size = 34, normalized size = 1. \[ \frac{f^{a-e} \log \left (d f^{2 b x+e}+c\right )}{2 b d \log (f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 47, normalized size = 1.4 \begin{align*}{\frac{{f}^{a}\ln \left ( c+d{{\rm e}^{-\ln \left ( f \right ) a+\ln \left ( f \right ) e}}{{\rm e}^{ \left ( 2\,bx+a \right ) \ln \left ( f \right ) }} \right ) }{2\,{f}^{e}d\ln \left ( f \right ) b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1564, size = 43, normalized size = 1.26 \begin{align*} \frac{f^{a - e} \log \left (d f^{2 \, b x + e} + c\right )}{2 \, b d \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78473, size = 72, normalized size = 2.12 \begin{align*} \frac{f^{a - e} \log \left (d f^{2 \, b x + e} + c\right )}{2 \, b d \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.914736, size = 42, normalized size = 1.24 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.47007, size = 50, normalized size = 1.47 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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