Optimal. Leaf size=42 \[ \frac{3 \left (a+b e^{2 x}\right )^{4/3}}{8 b^2}-\frac{3 a \sqrt [3]{a+b e^{2 x}}}{2 b^2} \]
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Rubi [A] time = 0.0478876, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2248, 43} \[ \frac{3 \left (a+b e^{2 x}\right )^{4/3}}{8 b^2}-\frac{3 a \sqrt [3]{a+b e^{2 x}}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{4 x}}{\left (a+b e^{2 x}\right )^{2/3}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{2/3}} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{2/3}}+\frac{\sqrt [3]{a+b x}}{b}\right ) \, dx,x,e^{2 x}\right )\\ &=-\frac{3 a \sqrt [3]{a+b e^{2 x}}}{2 b^2}+\frac{3 \left (a+b e^{2 x}\right )^{4/3}}{8 b^2}\\ \end{align*}
Mathematica [A] time = 0.0193891, size = 31, normalized size = 0.74 \[ \frac{3 \left (b e^{2 x}-3 a\right ) \sqrt [3]{a+b e^{2 x}}}{8 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 27, normalized size = 0.6 \begin{align*} -{\frac{-3\,b{{\rm e}^{2\,x}}+9\,a}{8\,{b}^{2}}\sqrt [3]{a+b{{\rm e}^{2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15377, size = 43, normalized size = 1.02 \begin{align*} \frac{3 \,{\left (b e^{\left (2 \, x\right )} + a\right )}^{\frac{4}{3}}}{8 \, b^{2}} - \frac{3 \,{\left (b e^{\left (2 \, x\right )} + a\right )}^{\frac{1}{3}} a}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44982, size = 66, normalized size = 1.57 \begin{align*} \frac{3 \,{\left (b e^{\left (2 \, x\right )} + a\right )}^{\frac{1}{3}}{\left (b e^{\left (2 \, x\right )} - 3 \, a\right )}}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{4 x}}{\left (a + b e^{2 x}\right )^{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26407, size = 39, normalized size = 0.93 \begin{align*} \frac{3 \,{\left ({\left (b e^{\left (2 \, x\right )} + a\right )}^{\frac{4}{3}} - 4 \,{\left (b e^{\left (2 \, x\right )} + a\right )}^{\frac{1}{3}} a\right )}}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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