Optimal. Leaf size=27 \[ -\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F)} \]
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Rubi [A] time = 0.042856, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2209} \[ -\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F)} \]
Antiderivative was successfully verified.
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Rule 2209
Rubi steps
\begin{align*} \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^3} \, dx &=-\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0093709, size = 27, normalized size = 1. \[ -\frac{F^{a+\frac{b}{(c+d x)^2}}}{2 b d \log (F)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 26, normalized size = 1. \begin{align*} -{\frac{1}{2\,\ln \left ( F \right ) bd}{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991408, size = 34, normalized size = 1.26 \begin{align*} -\frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{2 \, b d \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.57732, size = 115, normalized size = 4.26 \begin{align*} -\frac{F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, 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 = 0.440417, size = 54, normalized size = 2. \begin{align*} \begin{cases} - \frac{F^{a + \frac{b}{\left (c + d x\right )^{2}}}}{2 b d \log{\left (F \right )}} & \text{for}\: 2 b d \log{\left (F \right )} \neq 0 \\- \frac{1}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.183, size = 34, normalized size = 1.26 \begin{align*} -\frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{2 \, b d \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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