Optimal. Leaf size=71 \[ -\frac {\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (2+e x)^{7/2}}-\frac {\left (4-e^2 x^2\right )^{5/4}}{15\ 3^{3/4} e (2+e x)^{5/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 2, 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 {\left (4-e^2 x^2\right )^{5/4}}{15\ 3^{3/4} e (e x+2)^{5/2}}-\frac {\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} 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 {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx &=-\frac {\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (2+e x)^{7/2}}+\frac {1}{9} \int \frac {\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx\\ &=-\frac {\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (2+e x)^{7/2}}-\frac {\left (4-e^2 x^2\right )^{5/4}}{15\ 3^{3/4} e (2+e x)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 47, normalized size = 0.66 \begin {gather*} -\frac {(7+e x) \left (4 (2+e x)-(2+e x)^2\right )^{5/4}}{15\ 3^{3/4} e (2+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 35, normalized size = 0.49
method | result | size |
gosper | \(\frac {\left (e x -2\right ) \left (e x +7\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}}}{45 \left (e x +2\right )^{\frac {5}{2}} e}\) | \(35\) |
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 = 59, normalized size = 0.83 \begin {gather*} \frac {{\left (x^{2} e^{2} + 5 \, x e - 14\right )} {\left (-3 \, x^{2} e^{2} + 12\right )}^{\frac {1}{4}} \sqrt {x e + 2}}{45 \, {\left (x^{3} e^{4} + 6 \, x^{2} e^{3} + 12 \, x e^{2} + 8 \, 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*} \sqrt [4]{3} \int \frac {\sqrt [4]{- e^{2} x^{2} + 4}}{e^{3} x^{3} \sqrt {e x + 2} + 6 e^{2} x^{2} \sqrt {e x + 2} + 12 e x \sqrt {e x + 2} + 8 \sqrt {e x + 2}}\, dx \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 = 0.29, size = 37, normalized size = 0.52 \begin {gather*} \frac {{\left (12-3\,e^2\,x^2\right )}^{1/4}\,\left (e^2\,x^2+5\,e\,x-14\right )}{45\,e\,{\left (e\,x+2\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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