Optimal. Leaf size=35 \[ -\frac {\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (e x+2)^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {651} \begin {gather*} -\frac {\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (e x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin {align*} \int \frac {1}{(2+e x)^{3/2} \sqrt [4]{12-3 e^2 x^2}} \, dx &=-\frac {\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt [4]{3} e (2+e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 35, normalized size = 1.00 \begin {gather*} \frac {e x-2}{3 e \sqrt {e x+2} \sqrt [4]{12-3 e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 42, normalized size = 1.20 \begin {gather*} -\frac {\left (4 (e x+2)-(e x+2)^2\right )^{3/4}}{3 \sqrt [4]{3} e (e x+2)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 40, normalized size = 1.14 \begin {gather*} -\frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{9 \, {\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 30, normalized size = 0.86 \begin {gather*} \frac {e x -2}{3 \sqrt {e x +2}\, \left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} {\left (e x + 2\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 24, normalized size = 0.69 \begin {gather*} -\frac {{\left (12-3\,e^2\,x^2\right )}^{3/4}}{9\,e\,{\left (e\,x+2\right )}^{3/2}} \end {gather*}
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 \begin {gather*} \frac {3^{\frac {3}{4}} \int \frac {1}{e x \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4} + 2 \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4}}\, dx}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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