Optimal. Leaf size=27 \[ -\frac {1}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)} \]
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Rubi [A]
time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {2278, 32}
\begin {gather*} -\frac {1}{2 b d \log (F) \left (a+b F^{c+d x}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2278
Rubi steps
\begin {align*} \int \frac {F^{c+d x}}{\left (a+b F^{c+d x}\right )^3} \, dx &=\frac {\text {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*}
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Mathematica [A]
time = 0.04, size = 27, normalized size = 1.00 \begin {gather*} -\frac {1}{2 b d \left (a+b F^{c+d x}\right )^2 \log (F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 26, normalized size = 0.96
method | result | size |
derivativedivides | \(-\frac {1}{2 b d \left (a +b \,F^{d x +c}\right )^{2} \ln \left (F \right )}\) | \(26\) |
default | \(-\frac {1}{2 b d \left (a +b \,F^{d x +c}\right )^{2} \ln \left (F \right )}\) | \(26\) |
risch | \(-\frac {1}{2 b d \left (a +b \,F^{d x +c}\right )^{2} \ln \left (F \right )}\) | \(26\) |
norman | \(-\frac {1}{2 b \ln \left (F \right ) d \left (a +b \,{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}\right )^{2}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 25, normalized size = 0.93 \begin {gather*} -\frac {1}{2 \, {\left (F^{d x + c} b + a\right )}^{2} b d \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 46, normalized size = 1.70 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (22) = 44\).
time = 0.05, size = 53, normalized size = 1.96 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.19, size = 26, normalized size = 0.96 \begin {gather*} -\frac {1}{2 \, {\left (F^{d x} F^{c} b + a\right )}^{2} b d \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.63, size = 25, normalized size = 0.93 \begin {gather*} -\frac {1}{2\,b\,d\,\ln \left (F\right )\,{\left (a+F^{c+d\,x}\,b\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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