Optimal. Leaf size=28 \[ \frac {F^{-c} \log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]
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Rubi [A] time = 0.07, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2247, 2246, 31} \[ \frac {F^{-c} \log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2246
Rule 2247
Rubi steps
\begin {align*} \int \frac {F^{d x}}{a+b F^{c+d x}} \, dx &=F^{-c} \int \frac {F^{c+d x}}{a+b F^{c+d x}} \, dx\\ &=\frac {F^{-c} \operatorname {Subst}\left (\int \frac {1}{a+b x} \, dx,x,F^{c+d x}\right )}{d \log (F)}\\ &=\frac {F^{-c} \log \left (a+b F^{c+d x}\right )}{b d \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ \frac {F^{-c} \log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 28, normalized size = 1.00 \[ \frac {\log \left (F^{d x + c} b + a\right )}{F^{c} b d \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 30, normalized size = 1.07 \[ \frac {\log \left ({\left | F^{d x} F^{c} b + a \right |}\right )}{F^{c} b d \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 33, normalized size = 1.18 \[ \frac {F^{-c} \ln \left (b \,{\mathrm e}^{c \ln \relax (F )} {\mathrm e}^{d x \ln \relax (F )}+a \right )}{b d \ln \relax (F )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 28, normalized size = 1.00 \[ \frac {\log \left (F^{d x + c} b + a\right )}{F^{c} b d \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.58, size = 28, normalized size = 1.00 \[ \frac {\ln \left (a+F^{c+d\,x}\,b\right )}{F^c\,b\,d\,\ln \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 24, normalized size = 0.86 \[ \frac {e^{- c \log {\relax (F )}} \log {\left (F^{c + d x} + \frac {a}{b} \right )}}{b d \log {\relax (F )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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