Optimal. Leaf size=27 \[ -\frac{1}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]
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Rubi [A] time = 0.0345761, antiderivative size = 27, 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} \[ -\frac{1}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]
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 )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+b x)^3} \, dx,x,F^{c+d x}\right )}{d \log (F)}\\ &=-\frac{1}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0179545, size = 27, normalized size = 1. \[ -\frac{1}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 26, normalized size = 1. \begin{align*} -{\frac{1}{2\,bd \left ( a+b{F}^{dx+c} \right ) ^{2}\ln \left ( F \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02482, size = 34, normalized size = 1.26 \begin{align*} -\frac{1}{2 \,{\left (F^{d x + c} b + a\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 = 1.50665, size = 115, normalized size = 4.26 \begin{align*} -\frac{1}{2 \,{\left (2 \, F^{d x + c} a b^{2} d \log \left (F\right ) + F^{2 \, d x + 2 \, c} b^{3} d \log \left (F\right ) + a^{2} b d \log \left (F\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.137012, size = 53, normalized size = 1.96 \begin{align*} - \frac{1}{4 F^{c + d x} a b^{2} d \log{\left (F \right )} + 2 F^{2 c + 2 d x} b^{3} d \log{\left (F \right )} + 2 a^{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 = 1.28687, size = 34, normalized size = 1.26 \begin{align*} -\frac{1}{2 \,{\left (F^{d x + c} b + a\right )}^{2} b d \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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