Optimal. Leaf size=61 \[ \frac {f^{a+2 b x-e}}{2 b d \log (f)}-\frac {c f^{a-2 e} \log \left (d f^{2 b x+e}+c\right )}{2 b d^2 \log (f)} \]
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Rubi [A] time = 0.06, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2248, 43} \[ \frac {f^{a+2 b x-e}}{2 b d \log (f)}-\frac {c f^{a-2 e} \log \left (d f^{2 b x+e}+c\right )}{2 b d^2 \log (f)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2248
Rubi steps
\begin {align*} \int \frac {f^{a+4 b x}}{c+d f^{e+2 b x}} \, dx &=\frac {f^{a-2 e} \operatorname {Subst}\left (\int \frac {x}{c+d x} \, dx,x,f^{e+2 b x}\right )}{2 b \log (f)}\\ &=\frac {f^{a-2 e} \operatorname {Subst}\left (\int \left (\frac {1}{d}-\frac {c}{d (c+d x)}\right ) \, dx,x,f^{e+2 b x}\right )}{2 b \log (f)}\\ &=\frac {f^{a-e+2 b x}}{2 b d \log (f)}-\frac {c f^{a-2 e} \log \left (c+d f^{e+2 b x}\right )}{2 b d^2 \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 48, normalized size = 0.79 \[ \frac {f^{a-2 e} \left (d f^{2 b x+e}-c \log \left (d f^{2 b x+e}+c\right )\right )}{2 b d^2 \log (f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 53, normalized size = 0.87 \[ \frac {d f^{2 \, b x + e} f^{a - 2 \, e} - c f^{a - 2 \, e} \log \left (d f^{2 \, b x + e} + c\right )}{2 \, b d^{2} \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 66, normalized size = 1.08 \[ \frac {1}{2} \, f^{a} {\left (\frac {f^{2 \, b x}}{b d f^{e} \log \relax (f)} - \frac {c \log \left ({\left | d f^{2 \, b x} f^{e} + c \right |}\right )}{b d^{2} f^{2 \, e} \log \relax (f)}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 76, normalized size = 1.25 \[ -\frac {c \,f^{a} f^{-2 e} \ln \left (d \,{\mathrm e}^{\left (2 b x +e \right ) \ln \relax (f )}+c \right )}{2 b \,d^{2} \ln \relax (f )}+\frac {f^{a} f^{-2 e} {\mathrm e}^{\left (2 b x +e \right ) \ln \relax (f )}}{2 b d \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 65, normalized size = 1.07 \[ -\frac {c f^{a - 2 \, e} \log \left (d f^{2 \, b x + e} + c\right )}{2 \, b d^{2} \log \relax (f)} + \frac {{\left (d f^{2 \, b x + e} + c\right )} f^{a - 2 \, e}}{2 \, b d^{2} \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.54, size = 47, normalized size = 0.77 \[ -\frac {f^{a-2\,e}\,\left (\frac {c\,\ln \left (c+d\,f^{e+2\,b\,x}\right )}{2}-\frac {d\,f^{e+2\,b\,x}}{2}\right )}{b\,d^2\,\ln \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.19, size = 92, normalized size = 1.51 \[ \frac {\left (\begin {cases} x & \text {for}\: b = 0 \vee f = 1 \\\frac {e^{2 b x \log {\relax (f )}}}{2 b \log {\relax (f )}} & \text {otherwise} \end {cases}\right ) e^{a \log {\relax (f )}} e^{- e \log {\relax (f )}}}{d} - \frac {c e^{\left (a - 2 e\right ) \log {\relax (f )}} \log {\left (\frac {c e^{\frac {a \log {\relax (f )}}{2}} e^{- e \log {\relax (f )}}}{d} + \sqrt {e^{\left (a + 4 b x\right ) \log {\relax (f )}}} \right )}}{2 b d^{2} \log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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