Optimal. Leaf size=21 \[ \frac{e^{2 x}}{2 a \left (a+b e^x\right )^2} \]
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Rubi [A] time = 0.0226021, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2248, 37} \[ \frac{e^{2 x}}{2 a \left (a+b e^x\right )^2} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{2 x}}{\left (a+b e^x\right )^3} \, dx &=\operatorname{Subst}\left (\int \frac{x}{(a+b x)^3} \, dx,x,e^x\right )\\ &=\frac{e^{2 x}}{2 a \left (a+b e^x\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0092525, size = 21, normalized size = 1. \[ \frac{e^{2 x}}{2 a \left (a+b e^x\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 29, normalized size = 1.4 \begin{align*}{\frac{a}{2\,{b}^{2} \left ( a+b{{\rm e}^{x}} \right ) ^{2}}}-{\frac{1}{{b}^{2} \left ( a+b{{\rm e}^{x}} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11568, size = 82, normalized size = 3.9 \begin{align*} -\frac{b e^{x}}{b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}} - \frac{a}{2 \,{\left (b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.41622, size = 78, normalized size = 3.71 \begin{align*} -\frac{2 \, b e^{x} + a}{2 \,{\left (b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.123841, size = 37, normalized size = 1.76 \begin{align*} \frac{- a - 2 b e^{x}}{2 a^{2} b^{2} + 4 a b^{3} e^{x} + 2 b^{4} e^{2 x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24601, size = 27, normalized size = 1.29 \begin{align*} -\frac{2 \, b e^{x} + a}{2 \,{\left (b e^{x} + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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