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 = 0.05, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 43
Rule 2248
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.41, size = 25, normalized size = 0.60 \[ \frac {3 \, {\left (b e^{\left (2 \, x\right )} + a\right )}^{\frac {1}{3}} {\left (b e^{\left (2 \, x\right )} - 3 \, a\right )}}{8 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 32, normalized size = 0.76 \[ \frac {3 \, {\left (b e^{\left (2 \, x\right )} + a\right )}^{\frac {4}{3}}}{8 \, b^{2}} - \frac {3 \, {\left (b e^{\left (2 \, x\right )} + a\right )}^{\frac {1}{3}} a}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 27, normalized size = 0.64 \[ -\frac {3 \left (b \,{\mathrm e}^{2 x}+a \right )^{\frac {1}{3}} \left (-b \,{\mathrm e}^{2 x}+3 a \right )}{8 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 32, normalized size = 0.76 \[ \frac {3 \, {\left (b e^{\left (2 \, x\right )} + a\right )}^{\frac {4}{3}}}{8 \, b^{2}} - \frac {3 \, {\left (b e^{\left (2 \, x\right )} + a\right )}^{\frac {1}{3}} a}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.49, size = 26, normalized size = 0.62 \[ -\frac {3\,\left (3\,a-b\,{\mathrm {e}}^{2\,x}\right )\,{\left (a+b\,{\mathrm {e}}^{2\,x}\right )}^{1/3}}{8\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{4 x}}{\left (a + b e^{2 x}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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