Optimal. Leaf size=86 \[ -\frac {\sqrt {3} \sqrt {2-e x}}{64 e (e x+2)}-\frac {\sqrt {2-e x}}{8 \sqrt {3} e (e x+2)^2}-\frac {\sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{128 e} \]
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Rubi [A] time = 0.03, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {627, 51, 63, 206} \[ -\frac {\sqrt {3} \sqrt {2-e x}}{64 e (e x+2)}-\frac {\sqrt {2-e x}}{8 \sqrt {3} e (e x+2)^2}-\frac {\sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{128 e} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 627
Rubi steps
\begin {align*} \int \frac {1}{(2+e x)^{5/2} \sqrt {12-3 e^2 x^2}} \, dx &=\int \frac {1}{\sqrt {6-3 e x} (2+e x)^3} \, dx\\ &=-\frac {\sqrt {2-e x}}{8 \sqrt {3} e (2+e x)^2}+\frac {3}{16} \int \frac {1}{\sqrt {6-3 e x} (2+e x)^2} \, dx\\ &=-\frac {\sqrt {2-e x}}{8 \sqrt {3} e (2+e x)^2}-\frac {\sqrt {3} \sqrt {2-e x}}{64 e (2+e x)}+\frac {3}{128} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx\\ &=-\frac {\sqrt {2-e x}}{8 \sqrt {3} e (2+e x)^2}-\frac {\sqrt {3} \sqrt {2-e x}}{64 e (2+e x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{64 e}\\ &=-\frac {\sqrt {2-e x}}{8 \sqrt {3} e (2+e x)^2}-\frac {\sqrt {3} \sqrt {2-e x}}{64 e (2+e x)}-\frac {\sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{128 e}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 53, normalized size = 0.62 \[ \frac {(e x-2) \sqrt {e x+2} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {1}{2}-\frac {e x}{4}\right )}{32 e \sqrt {12-3 e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 139, normalized size = 1.62 \[ \frac {3 \, \sqrt {3} {\left (e^{3} x^{3} + 6 \, e^{2} x^{2} + 12 \, e x + 8\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (3 \, e x + 14\right )} \sqrt {e x + 2}}{768 \, {\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 126, normalized size = 1.47 \[ -\frac {\sqrt {-e^{2} x^{2}+4}\, \left (3 \sqrt {3}\, e^{2} x^{2} \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+12 \sqrt {3}\, e x \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+6 \sqrt {-3 e x +6}\, e x +12 \sqrt {3}\, \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+28 \sqrt {-3 e x +6}\right ) \sqrt {3}}{384 \sqrt {\left (e x +2\right )^{5}}\, \sqrt {-3 e x +6}\, e} \]
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 \[ \int \frac {1}{\sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {12-3\,e^2\,x^2}\,{\left (e\,x+2\right )}^{5/2}} \,d x \]
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 \[ \frac {\sqrt {3} \int \frac {1}{e^{2} x^{2} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} + 4 e x \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}}\, dx}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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