Optimal. Leaf size=113 \[ -\frac {\sqrt {3} (2-e x)^{3/2}}{e (e x+2)^3}-\frac {3 \sqrt {3} \sqrt {2-e x}}{32 e (e x+2)}+\frac {3 \sqrt {3} \sqrt {2-e x}}{4 e (e x+2)^2}-\frac {3 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{64 e} \]
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Rubi [A] time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {627, 47, 51, 63, 206} \begin {gather*} -\frac {\sqrt {3} (2-e x)^{3/2}}{e (e x+2)^3}-\frac {3 \sqrt {3} \sqrt {2-e x}}{32 e (e x+2)}+\frac {3 \sqrt {3} \sqrt {2-e x}}{4 e (e x+2)^2}-\frac {3 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{64 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 206
Rule 627
Rubi steps
\begin {align*} \int \frac {\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{11/2}} \, dx &=\int \frac {(6-3 e x)^{3/2}}{(2+e x)^4} \, dx\\ &=-\frac {\sqrt {3} (2-e x)^{3/2}}{e (2+e x)^3}-\frac {3}{2} \int \frac {\sqrt {6-3 e x}}{(2+e x)^3} \, dx\\ &=-\frac {\sqrt {3} (2-e x)^{3/2}}{e (2+e x)^3}+\frac {3 \sqrt {3} \sqrt {2-e x}}{4 e (2+e x)^2}+\frac {9}{8} \int \frac {1}{\sqrt {6-3 e x} (2+e x)^2} \, dx\\ &=-\frac {\sqrt {3} (2-e x)^{3/2}}{e (2+e x)^3}+\frac {3 \sqrt {3} \sqrt {2-e x}}{4 e (2+e x)^2}-\frac {3 \sqrt {3} \sqrt {2-e x}}{32 e (2+e x)}+\frac {9}{64} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx\\ &=-\frac {\sqrt {3} (2-e x)^{3/2}}{e (2+e x)^3}+\frac {3 \sqrt {3} \sqrt {2-e x}}{4 e (2+e x)^2}-\frac {3 \sqrt {3} \sqrt {2-e x}}{32 e (2+e x)}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{32 e}\\ &=-\frac {\sqrt {3} (2-e x)^{3/2}}{e (2+e x)^3}+\frac {3 \sqrt {3} \sqrt {2-e x}}{4 e (2+e x)^2}-\frac {3 \sqrt {3} \sqrt {2-e x}}{32 e (2+e x)}-\frac {3 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{64 e}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 55, normalized size = 0.49 \begin {gather*} -\frac {3 (e x-2)^2 \sqrt {12-3 e^2 x^2} \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {1}{2}-\frac {e x}{4}\right )}{640 e \sqrt {e x+2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.44, size = 106, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {3} \sqrt {4 (e x+2)-(e x+2)^2} \left (3 (e x+2)^2-56 (e x+2)+128\right )}{32 e (e x+2)^{7/2}}-\frac {3 \sqrt {3} \tanh ^{-1}\left (\frac {2 \sqrt {e x+2}}{\sqrt {4 (e x+2)-(e x+2)^2}}\right )}{64 e} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 163, normalized size = 1.44 \begin {gather*} \frac {3 \, \sqrt {3} {\left (e^{4} x^{4} + 8 \, e^{3} x^{3} + 24 \, e^{2} x^{2} + 32 \, 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 (3 \, e^{2} x^{2} - 44 \, e x + 28\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{128 \, {\left (e^{5} x^{4} + 8 \, e^{4} x^{3} + 24 \, e^{3} x^{2} + 32 \, 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 = 0.10, size = 167, normalized size = 1.48 \begin {gather*} -\frac {\sqrt {-e^{2} x^{2}+4}\, \left (3 \sqrt {3}\, e^{3} x^{3} \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+18 \sqrt {3}\, e^{2} x^{2} \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+6 \sqrt {-3 e x +6}\, e^{2} x^{2}+36 \sqrt {3}\, e x \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )-88 \sqrt {-3 e x +6}\, e x +24 \sqrt {3}\, \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+56 \sqrt {-3 e x +6}\right ) \sqrt {3}}{64 \sqrt {\left (e x +2\right )^{7}}\, \sqrt {-3 e x +6}\, 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 {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}}}{{\left (e x + 2\right )}^{\frac {11}{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 {{\left (12-3\,e^2\,x^2\right )}^{3/2}}{{\left (e\,x+2\right )}^{11/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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