Optimal. Leaf size=141 \[ -\frac {\left (4-e^2 x^2\right )^{3/4}}{15 \sqrt [4]{3} e (2+e x)^{9/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{55 \sqrt [4]{3} e (2+e x)^{7/2}}-\frac {2 \left (4-e^2 x^2\right )^{3/4}}{385 \sqrt [4]{3} e (2+e x)^{5/2}}-\frac {2 \left (4-e^2 x^2\right )^{3/4}}{1155 \sqrt [4]{3} e (2+e x)^{3/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 4, 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}}{1155 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac {2 \left (4-e^2 x^2\right )^{3/4}}{385 \sqrt [4]{3} e (e x+2)^{5/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{55 \sqrt [4]{3} e (e x+2)^{7/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{15 \sqrt [4]{3} e (e x+2)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 665
Rule 673
Rubi steps
\begin {align*} \int \frac {1}{(2+e x)^{9/2} \sqrt [4]{12-3 e^2 x^2}} \, dx &=-\frac {\left (4-e^2 x^2\right )^{3/4}}{15 \sqrt [4]{3} e (2+e x)^{9/2}}+\frac {1}{5} \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}}{15 \sqrt [4]{3} e (2+e x)^{9/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{55 \sqrt [4]{3} e (2+e x)^{7/2}}+\frac {2}{55} \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}}{15 \sqrt [4]{3} e (2+e x)^{9/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{55 \sqrt [4]{3} e (2+e x)^{7/2}}-\frac {2 \left (4-e^2 x^2\right )^{3/4}}{385 \sqrt [4]{3} e (2+e x)^{5/2}}+\frac {2}{385} \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}}{15 \sqrt [4]{3} e (2+e x)^{9/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{55 \sqrt [4]{3} e (2+e x)^{7/2}}-\frac {2 \left (4-e^2 x^2\right )^{3/4}}{385 \sqrt [4]{3} e (2+e x)^{5/2}}-\frac {2 \left (4-e^2 x^2\right )^{3/4}}{1155 \sqrt [4]{3} e (2+e x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 57, normalized size = 0.40 \begin {gather*} -\frac {\left (4-e^2 x^2\right )^{3/4} \left (159+69 e x+18 e^2 x^2+2 e^3 x^3\right )}{1155 \sqrt [4]{3} e (2+e x)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 52, normalized size = 0.37
method | result | size |
gosper | \(\frac {\left (e x -2\right ) \left (2 e^{3} x^{3}+18 e^{2} x^{2}+69 e x +159\right )}{1155 \left (e x +2\right )^{\frac {7}{2}} e \left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}}}\) | \(52\) |
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 = 3.84, size = 81, normalized size = 0.57 \begin {gather*} -\frac {{\left (2 \, x^{3} e^{3} + 18 \, x^{2} e^{2} + 69 \, x e + 159\right )} {\left (-3 \, x^{2} e^{2} + 12\right )}^{\frac {3}{4}} \sqrt {x e + 2}}{3465 \, {\left (x^{5} e^{6} + 10 \, x^{4} e^{5} + 40 \, x^{3} e^{4} + 80 \, x^{2} e^{3} + 80 \, x e^{2} + 32 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.05, size = 72, normalized size = 0.51 \begin {gather*} -\frac {1}{221760} \cdot 3^{\frac {3}{4}} {\left (77 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {15}{4}} + 315 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {11}{4}} + 495 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {7}{4}} + 385 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {3}{4}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 111, normalized size = 0.79 \begin {gather*} -\frac {{\left (12-3\,e^2\,x^2\right )}^{3/4}\,\left (\frac {23\,x}{1155\,e^4}+\frac {53}{1155\,e^5}+\frac {2\,x^3}{3465\,e^2}+\frac {2\,x^2}{385\,e^3}\right )}{\frac {16\,\sqrt {e\,x+2}}{e^4}+x^4\,\sqrt {e\,x+2}+\frac {32\,x\,\sqrt {e\,x+2}}{e^3}+\frac {8\,x^3\,\sqrt {e\,x+2}}{e}+\frac {24\,x^2\,\sqrt {e\,x+2}}{e^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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