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 = 0.04, antiderivative size = 25, 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 = {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 32
Rule 2246
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*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.00 \[ -\frac {1}{b d \log (F) \left (a+b F^{c+d x}\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 25, normalized size = 1.00 \[ -\frac {1}{F^{d x + c} b^{2} d \log \relax (F) + a b d \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 26, normalized size = 1.04 \[ -\frac {1}{{\left (F^{d x} F^{c} b + a\right )} b d \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 1.04 \[ -\frac {1}{\left (b \,F^{d x +c}+a \right ) b d \ln \relax (F )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 25, normalized size = 1.00 \[ -\frac {1}{{\left (F^{d x + c} b + a\right )} b d \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.46, size = 31, normalized size = 1.24 \[ \frac {F^{c+d\,x}}{a^2\,d\,\ln \relax (F)+F^{c+d\,x}\,a\,b\,d\,\ln \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 26, normalized size = 1.04 \[ - \frac {1}{F^{c + d x} b^{2} d \log {\relax (F )} + a b d \log {\relax (F )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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