Optimal. Leaf size=34 \[ \frac{a}{3 b^2 \left (a+b e^x\right )^3}-\frac{1}{2 b^2 \left (a+b e^x\right )^2} \]
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Rubi [A] time = 0.0359505, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2248, 43} \[ \frac{a}{3 b^2 \left (a+b e^x\right )^3}-\frac{1}{2 b^2 \left (a+b e^x\right )^2} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 x}}{\left (a+b e^x\right )^4} \, dx &=\operatorname{Subst}\left (\int \frac{x}{(a+b x)^4} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^4}+\frac{1}{b (a+b x)^3}\right ) \, dx,x,e^x\right )\\ &=\frac{a}{3 b^2 \left (a+b e^x\right )^3}-\frac{1}{2 b^2 \left (a+b e^x\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0211904, size = 24, normalized size = 0.71 \[ -\frac{a+3 b e^x}{6 b^2 \left (a+b e^x\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 29, normalized size = 0.9 \begin{align*}{\frac{a}{3\,{b}^{2} \left ( a+b{{\rm e}^{x}} \right ) ^{3}}}-{\frac{1}{2\,{b}^{2} \left ( a+b{{\rm e}^{x}} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08893, size = 115, normalized size = 3.38 \begin{align*} -\frac{b e^{x}}{2 \,{\left (b^{5} e^{\left (3 \, x\right )} + 3 \, a b^{4} e^{\left (2 \, x\right )} + 3 \, a^{2} b^{3} e^{x} + a^{3} b^{2}\right )}} - \frac{a}{6 \,{\left (b^{5} e^{\left (3 \, x\right )} + 3 \, a b^{4} e^{\left (2 \, x\right )} + 3 \, a^{2} b^{3} e^{x} + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52872, size = 105, normalized size = 3.09 \begin{align*} -\frac{3 \, b e^{x} + a}{6 \,{\left (b^{5} e^{\left (3 \, x\right )} + 3 \, a b^{4} e^{\left (2 \, x\right )} + 3 \, a^{2} b^{3} e^{x} + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.146506, size = 51, normalized size = 1.5 \begin{align*} \frac{- a - 3 b e^{x}}{6 a^{3} b^{2} + 18 a^{2} b^{3} e^{x} + 18 a b^{4} e^{2 x} + 6 b^{5} e^{3 x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20175, size = 27, normalized size = 0.79 \begin{align*} -\frac{3 \, b e^{x} + a}{6 \,{\left (b e^{x} + a\right )}^{3} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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