Optimal. Leaf size=108 \[ \frac {5}{256 \sqrt {3} e \sqrt {2-e x}}-\frac {5}{192 \sqrt {3} e \sqrt {2-e x} (e x+2)}-\frac {1}{24 \sqrt {3} e \sqrt {2-e x} (e x+2)^2}-\frac {5 \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{512 \sqrt {3} e} \]
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Rubi [A] time = 0.04, antiderivative size = 115, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {627, 51, 63, 206} \begin {gather*} -\frac {5 \sqrt {2-e x}}{256 \sqrt {3} e (e x+2)}-\frac {5 \sqrt {2-e x}}{96 \sqrt {3} e (e x+2)^2}+\frac {1}{6 \sqrt {3} e \sqrt {2-e x} (e x+2)^2}-\frac {5 \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{512 \sqrt {3} e} \end {gather*}
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)^{3/2} \left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac {1}{(6-3 e x)^{3/2} (2+e x)^3} \, dx\\ &=\frac {1}{6 \sqrt {3} e \sqrt {2-e x} (2+e x)^2}+\frac {5}{12} \int \frac {1}{\sqrt {6-3 e x} (2+e x)^3} \, dx\\ &=\frac {1}{6 \sqrt {3} e \sqrt {2-e x} (2+e x)^2}-\frac {5 \sqrt {2-e x}}{96 \sqrt {3} e (2+e x)^2}+\frac {5}{64} \int \frac {1}{\sqrt {6-3 e x} (2+e x)^2} \, dx\\ &=\frac {1}{6 \sqrt {3} e \sqrt {2-e x} (2+e x)^2}-\frac {5 \sqrt {2-e x}}{96 \sqrt {3} e (2+e x)^2}-\frac {5 \sqrt {2-e x}}{256 \sqrt {3} e (2+e x)}+\frac {5}{512} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx\\ &=\frac {1}{6 \sqrt {3} e \sqrt {2-e x} (2+e x)^2}-\frac {5 \sqrt {2-e x}}{96 \sqrt {3} e (2+e x)^2}-\frac {5 \sqrt {2-e x}}{256 \sqrt {3} e (2+e x)}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{768 e}\\ &=\frac {1}{6 \sqrt {3} e \sqrt {2-e x} (2+e x)^2}-\frac {5 \sqrt {2-e x}}{96 \sqrt {3} e (2+e x)^2}-\frac {5 \sqrt {2-e x}}{256 \sqrt {3} e (2+e x)}-\frac {5 \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{512 \sqrt {3} e}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 48, normalized size = 0.44 \begin {gather*} \frac {\sqrt {e x+2} \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};\frac {1}{2}-\frac {e x}{4}\right )}{96 e \sqrt {12-3 e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.49, size = 113, normalized size = 1.05 \begin {gather*} -\frac {\sqrt {4 (e x+2)-(e x+2)^2} \left (15 (e x+2)^2-20 (e x+2)-32\right )}{768 \sqrt {3} e (e x-2) (e x+2)^{5/2}}-\frac {5 \tanh ^{-1}\left (\frac {2 \sqrt {e x+2}}{\sqrt {4 (e x+2)-(e x+2)^2}}\right )}{512 \sqrt {3} e} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 147, normalized size = 1.36 \begin {gather*} \frac {15 \, \sqrt {3} {\left (e^{4} x^{4} + 4 \, e^{3} x^{3} - 16 \, e x - 16\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 \, {\left (15 \, e^{2} x^{2} + 40 \, e x - 12\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{9216 \, {\left (e^{5} x^{4} + 4 \, e^{4} x^{3} - 16 \, e^{2} x - 16 \, e\right )}} \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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maple [A] time = 0.07, size = 135, normalized size = 1.25 \begin {gather*} \frac {\sqrt {-3 e^{2} x^{2}+12}\, \left (5 \sqrt {3}\, \sqrt {-3 e x +6}\, e^{2} x^{2} \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )-30 e^{2} x^{2}+20 \sqrt {3}\, \sqrt {-3 e x +6}\, e x \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )-80 e x +20 \sqrt {3}\, \sqrt {-3 e x +6}\, \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+24\right )}{4608 \left (e x +2\right )^{\frac {5}{2}} \left (e x -2\right ) e} \end {gather*}
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*} \int \frac {1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}} {\left (e x + 2\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (12-3\,e^2\,x^2\right )}^{3/2}\,{\left (e\,x+2\right )}^{3/2}} \,d x \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 {\sqrt {3} \int \frac {1}{- e^{3} x^{3} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} - 2 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} + 8 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}}\, dx}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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