Optimal. Leaf size=144 \[ -\frac {3 \sqrt {3} (2-e x)^{3/2}}{4 e (e x+2)^4}-\frac {9 \sqrt {3} \sqrt {2-e x}}{1024 e (e x+2)}-\frac {3 \sqrt {3} \sqrt {2-e x}}{128 e (e x+2)^2}+\frac {3 \sqrt {3} \sqrt {2-e x}}{8 e (e x+2)^3}-\frac {9 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{2048 e} \]
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Rubi [A] time = 0.05, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {3 \sqrt {3} (2-e x)^{3/2}}{4 e (e x+2)^4}-\frac {9 \sqrt {3} \sqrt {2-e x}}{1024 e (e x+2)}-\frac {3 \sqrt {3} \sqrt {2-e x}}{128 e (e x+2)^2}+\frac {3 \sqrt {3} \sqrt {2-e x}}{8 e (e x+2)^3}-\frac {9 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{2048 e} \]
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)^{13/2}} \, dx &=\int \frac {(6-3 e x)^{3/2}}{(2+e x)^5} \, dx\\ &=-\frac {3 \sqrt {3} (2-e x)^{3/2}}{4 e (2+e x)^4}-\frac {9}{8} \int \frac {\sqrt {6-3 e x}}{(2+e x)^4} \, dx\\ &=-\frac {3 \sqrt {3} (2-e x)^{3/2}}{4 e (2+e x)^4}+\frac {3 \sqrt {3} \sqrt {2-e x}}{8 e (2+e x)^3}+\frac {9}{16} \int \frac {1}{\sqrt {6-3 e x} (2+e x)^3} \, dx\\ &=-\frac {3 \sqrt {3} (2-e x)^{3/2}}{4 e (2+e x)^4}+\frac {3 \sqrt {3} \sqrt {2-e x}}{8 e (2+e x)^3}-\frac {3 \sqrt {3} \sqrt {2-e x}}{128 e (2+e x)^2}+\frac {27}{256} \int \frac {1}{\sqrt {6-3 e x} (2+e x)^2} \, dx\\ &=-\frac {3 \sqrt {3} (2-e x)^{3/2}}{4 e (2+e x)^4}+\frac {3 \sqrt {3} \sqrt {2-e x}}{8 e (2+e x)^3}-\frac {3 \sqrt {3} \sqrt {2-e x}}{128 e (2+e x)^2}-\frac {9 \sqrt {3} \sqrt {2-e x}}{1024 e (2+e x)}+\frac {27 \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx}{2048}\\ &=-\frac {3 \sqrt {3} (2-e x)^{3/2}}{4 e (2+e x)^4}+\frac {3 \sqrt {3} \sqrt {2-e x}}{8 e (2+e x)^3}-\frac {3 \sqrt {3} \sqrt {2-e x}}{128 e (2+e x)^2}-\frac {9 \sqrt {3} \sqrt {2-e x}}{1024 e (2+e x)}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{1024 e}\\ &=-\frac {3 \sqrt {3} (2-e x)^{3/2}}{4 e (2+e x)^4}+\frac {3 \sqrt {3} \sqrt {2-e x}}{8 e (2+e x)^3}-\frac {3 \sqrt {3} \sqrt {2-e x}}{128 e (2+e x)^2}-\frac {9 \sqrt {3} \sqrt {2-e x}}{1024 e (2+e x)}-\frac {9 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{2048 e}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 55, normalized size = 0.38 \[ -\frac {3 (e x-2)^2 \sqrt {12-3 e^2 x^2} \, _2F_1\left (\frac {5}{2},5;\frac {7}{2};\frac {1}{2}-\frac {e x}{4}\right )}{2560 e \sqrt {e x+2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 187, normalized size = 1.30 \[ \frac {3 \, {\left (3 \, \sqrt {3} {\left (e^{5} x^{5} + 10 \, e^{4} x^{4} + 40 \, e^{3} x^{3} + 80 \, e^{2} x^{2} + 80 \, e x + 32\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, {\left (3 \, e^{3} x^{3} + 26 \, e^{2} x^{2} - 316 \, e x + 312\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}\right )}}{4096 \, {\left (e^{6} x^{5} + 10 \, e^{5} x^{4} + 40 \, e^{4} x^{3} + 80 \, e^{3} x^{2} + 80 \, e^{2} x + 32 \, e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 208, normalized size = 1.44 \[ -\frac {3 \sqrt {-e^{2} x^{2}+4}\, \left (3 \sqrt {3}\, e^{4} x^{4} \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+24 \sqrt {3}\, e^{3} x^{3} \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+6 \sqrt {-3 e x +6}\, e^{3} x^{3}+72 \sqrt {3}\, e^{2} x^{2} \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+52 \sqrt {-3 e x +6}\, e^{2} x^{2}+96 \sqrt {3}\, e x \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )-632 \sqrt {-3 e x +6}\, e x +48 \sqrt {3}\, \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )+624 \sqrt {-3 e x +6}\right ) \sqrt {3}}{2048 \sqrt {\left (e x +2\right )^{9}}\, \sqrt {-3 e x +6}\, e} \]
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 \[ \int \frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{2}}}{{\left (e x + 2\right )}^{\frac {13}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (12-3\,e^2\,x^2\right )}^{3/2}}{{\left (e\,x+2\right )}^{13/2}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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