Optimal. Leaf size=25 \[ -\frac{1}{b d \log (F) \left (a+b F^{c+d x}\right )} \]
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Rubi [A] time = 0.0352445, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2246, 32} \[ -\frac{1}{b d \log (F) \left (a+b F^{c+d x}\right )} \]
Antiderivative was successfully verified.
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Rule 2246
Rule 32
Rubi steps
\begin{align*} \int \frac{F^{c+d x}}{\left (a+b F^{c+d x}\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+b x)^2} \, dx,x,F^{c+d x}\right )}{d \log (F)}\\ &=-\frac{1}{b d \left (a+b F^{c+d x}\right ) \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0198392, size = 25, normalized size = 1. \[ -\frac{1}{b d \log (F) \left (a+b F^{c+d x}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 1. \begin{align*} -{\frac{1}{bd \left ( a+b{F}^{dx+c} \right ) \ln \left ( F \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1812, size = 34, normalized size = 1.36 \begin{align*} -\frac{1}{{\left (F^{d x + c} b + a\right )} 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 = 1.47666, size = 62, normalized size = 2.48 \begin{align*} -\frac{1}{F^{d x + c} b^{2} d \log \left (F\right ) + a 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 = 0.115061, size = 26, normalized size = 1.04 \begin{align*} - \frac{1}{F^{c + d x} b^{2} d \log{\left (F \right )} + a 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 = 1.27317, size = 34, normalized size = 1.36 \begin{align*} -\frac{1}{{\left (F^{d x + c} b + a\right )} b d \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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