Optimal. Leaf size=71 \[ -\frac {\left (4-e^2 x^2\right )^{3/4}}{7 \sqrt [4]{3} e (2+e x)^{5/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{21 \sqrt [4]{3} e (2+e x)^{3/2}} \]
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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 )^{3/4}}{21 \sqrt [4]{3} e (e x+2)^{3/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{7 \sqrt [4]{3} e (e x+2)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 665
Rule 673
Rubi steps
\begin {align*} \int \frac {1}{(2+e x)^{5/2} \sqrt [4]{12-3 e^2 x^2}} \, dx &=-\frac {\left (4-e^2 x^2\right )^{3/4}}{7 \sqrt [4]{3} e (2+e x)^{5/2}}+\frac {1}{7} \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}}{7 \sqrt [4]{3} e (2+e x)^{5/2}}-\frac {\left (4-e^2 x^2\right )^{3/4}}{21 \sqrt [4]{3} e (2+e x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 40, normalized size = 0.56 \begin {gather*} -\frac {(5+e x) \left (4-e^2 x^2\right )^{3/4}}{21 \sqrt [4]{3} e (2+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.55, size = 35, normalized size = 0.49
method | result | size |
gosper | \(\frac {\left (e x -2\right ) \left (e x +5\right )}{21 \left (e x +2\right )^{\frac {3}{2}} e \left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}}}\) | \(35\) |
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 = 2.97, size = 52, normalized size = 0.73 \begin {gather*} -\frac {{\left (-3 \, x^{2} e^{2} + 12\right )}^{\frac {3}{4}} {\left (x e + 5\right )} \sqrt {x e + 2}}{63 \, {\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*} \frac {3^{\frac {3}{4}} \int \frac {1}{e^{2} x^{2} \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4} + 4 e x \sqrt {e x + 2} \sqrt [4]{- e^{2} x^{2} + 4} + 4 \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 [A]
time = 1.25, size = 40, normalized size = 0.56 \begin {gather*} -\frac {1}{252} \cdot 3^{\frac {3}{4}} {\left (3 \, {\left (\frac {4}{x e + 2} - 1\right )}^{\frac {7}{4}} + 7 \, {\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 = 0.68, size = 65, normalized size = 0.92 \begin {gather*} -\frac {\left (\frac {x}{63\,e^2}+\frac {5}{63\,e^3}\right )\,{\left (12-3\,e^2\,x^2\right )}^{3/4}}{\frac {4\,\sqrt {e\,x+2}}{e^2}+x^2\,\sqrt {e\,x+2}+\frac {4\,x\,\sqrt {e\,x+2}}{e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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