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 = 0.0707328, antiderivative size = 28, normalized size of antiderivative = 1., 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 2247
Rule 2246
Rule 31
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*}
Mathematica [A] time = 0.0089136, size = 28, normalized size = 1. \[ \frac{F^{-c} \log \left (a+b F^{c+d x}\right )}{b d \log (F)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 33, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( a+b{{\rm e}^{c\ln \left ( F \right ) }}{{\rm e}^{d\ln \left ( F \right ) x}} \right ) }{{F}^{c}b\ln \left ( F \right ) d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06323, size = 38, normalized size = 1.36 \begin{align*} \frac{\log \left (F^{d x + c} b + a\right )}{F^{c} 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 = 1.47925, size = 55, normalized size = 1.96 \begin{align*} \frac{\log \left (F^{d x + c} b + a\right )}{F^{c} 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 = 0.516429, size = 24, normalized size = 0.86 \begin{align*} \frac{e^{- c \log{\left (F \right )}} \log{\left (F^{c + d x} + \frac{a}{b} \right )}}{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 = 1.27375, size = 41, normalized size = 1.46 \begin{align*} \frac{\log \left ({\left | F^{d x} F^{c} b + a \right |}\right )}{F^{c} b d \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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