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 = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ -\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 54, normalized size = 2.00 \[ -\frac {F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, b d \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 54, normalized size = 2.00 \[ -\frac {F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, b d \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.96 \[ -\frac {F^{a +\frac {b}{\left (d x +c \right )^{2}}}}{2 b d \ln \relax (F )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 25, normalized size = 0.93 \[ -\frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{2 \, b d \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.54, size = 37, normalized size = 1.37 \[ -\frac {F^a\,F^{\frac {b}{c^2+2\,c\,d\,x+d^2\,x^2}}}{2\,b\,d\,\ln \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 54, normalized size = 2.00 \[ \begin {cases} - \frac {F^{a + \frac {b}{\left (c + d x\right )^{2}}}}{2 b d \log {\relax (F )}} & \text {for}\: 2 b d \log {\relax (F )} \neq 0 \\- \frac {1}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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