Optimal. Leaf size=106 \[ -\frac {\left (4-e^2 x^2\right )^{3/4}}{11 \sqrt [4]{3} e (2+e x)^{7/2}}-\frac {2 \left (4-e^2 x^2\right )^{3/4}}{77 \sqrt [4]{3} e (2+e x)^{5/2}}-\frac {2 \left (4-e^2 x^2\right )^{3/4}}{231 \sqrt [4]{3} e (2+e x)^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665}
\begin {gather*} -\frac {2 \left (4-e^2 x^2\right )^{3/4}}{231 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac {2 \left (4-e^2 x^2\right )^{3/4}}{77 \sqrt [4]{3} e (e x+2)^{5/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{11 \sqrt [4]{3} e (e x+2)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 665
Rule 673
Rubi steps
\begin {align*} \int \frac {1}{(2+e x)^{7/2} \sqrt [4]{12-3 e^2 x^2}} \, dx &=-\frac {\left (4-e^2 x^2\right )^{3/4}}{11 \sqrt [4]{3} e (2+e x)^{7/2}}+\frac {2}{11} \int \frac {1}{(2+e x)^{5/2} \sqrt [4]{12-3 e^2 x^2}} \, dx\\ &=-\frac {\left (4-e^2 x^2\right )^{3/4}}{11 \sqrt [4]{3} e (2+e x)^{7/2}}-\frac {2 \left (4-e^2 x^2\right )^{3/4}}{77 \sqrt [4]{3} e (2+e x)^{5/2}}+\frac {2}{77} \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}}{11 \sqrt [4]{3} e (2+e x)^{7/2}}-\frac {2 \left (4-e^2 x^2\right )^{3/4}}{77 \sqrt [4]{3} e (2+e x)^{5/2}}-\frac {2 \left (4-e^2 x^2\right )^{3/4}}{231 \sqrt [4]{3} e (2+e x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 49, normalized size = 0.46 \begin {gather*} -\frac {\left (4-e^2 x^2\right )^{3/4} \left (41+14 e x+2 e^2 x^2\right )}{231 \sqrt [4]{3} e (2+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 44, normalized size = 0.42
method | result | size |
gosper | \(\frac {\left (e x -2\right ) \left (2 e^{2} x^{2}+14 e x +41\right )}{231 \left (e x +2\right )^{\frac {5}{2}} e \left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}}}\) | \(44\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.91, size = 67, normalized size = 0.63 \begin {gather*} -\frac {{\left (2 \, x^{2} e^{2} + 14 \, x e + 41\right )} {\left (-3 \, x^{2} e^{2} + 12\right )}^{\frac {3}{4}} \sqrt {x e + 2}}{693 \, {\left (x^{4} e^{5} + 8 \, x^{3} e^{4} + 24 \, x^{2} e^{3} + 32 \, x e^{2} + 16 \, e\right )}} \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^{3} x^{3} \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4} + 6 e^{2} x^{2} \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4} + 12 e x \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4} + 8 \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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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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Mupad [B]
time = 0.34, size = 88, normalized size = 0.83 \begin {gather*} -\frac {{\left (12-3\,e^2\,x^2\right )}^{3/4}\,\left (\frac {2\,x}{99\,e^3}+\frac {41}{693\,e^4}+\frac {2\,x^2}{693\,e^2}\right )}{\frac {8\,\sqrt {e\,x+2}}{e^3}+x^3\,\sqrt {e\,x+2}+\frac {12\,x\,\sqrt {e\,x+2}}{e^2}+\frac {6\,x^2\,\sqrt {e\,x+2}}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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