Optimal. Leaf size=144 \[ -\frac {3 \sqrt {3} (2-e x)^{3/2}}{4 e (2+e x)^4}+\frac {3 \sqrt {3} \sqrt {2-e x}}{8 e (2+e x)^3}-\frac {3 \sqrt {3} \sqrt {2-e x}}{128 e (2+e x)^2}-\frac {9 \sqrt {3} \sqrt {2-e x}}{1024 e (2+e x)}-\frac {9 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{2048 e} \]
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Rubi [A]
time = 0.04, 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 = {641, 43, 44, 65,
212} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 212
Rule 641
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 \text {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 [A]
time = 0.36, size = 88, normalized size = 0.61 \begin {gather*} \frac {3 \sqrt {3} \left (-\frac {2 \sqrt {4-e^2 x^2} \left (312-316 e x+26 e^2 x^2+3 e^3 x^3\right )}{(2+e x)^{9/2}}-3 \tanh ^{-1}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )\right )}{2048 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 208, normalized size = 1.44
method | result | size |
default | \(-\frac {3 \sqrt {-e^{2} x^{2}+4}\, \left (3 \sqrt {3}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e^{4} x^{4}+24 \sqrt {3}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e^{3} x^{3}+6 e^{3} x^{3} \sqrt {-3 e x +6}+72 \sqrt {3}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e^{2} x^{2}+52 e^{2} x^{2} \sqrt {-3 e x +6}+96 \sqrt {3}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e x -632 e x \sqrt {-3 e x +6}+48 \sqrt {3}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )+624 \sqrt {-3 e x +6}\right ) \sqrt {3}}{2048 \sqrt {\left (e x +2\right )^{9}}\, \sqrt {-3 e x +6}\, e}\) | \(208\) |
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 = 4.03, size = 179, normalized size = 1.24 \begin {gather*} \frac {3 \, {\left (3 \, \sqrt {3} {\left (x^{5} e^{5} + 10 \, x^{4} e^{4} + 40 \, x^{3} e^{3} + 80 \, x^{2} e^{2} + 80 \, x e + 32\right )} \log \left (-\frac {3 \, x^{2} e^{2} - 12 \, x e + 4 \, \sqrt {3} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2} - 36}{x^{2} e^{2} + 4 \, x e + 4}\right ) - 4 \, {\left (3 \, x^{3} e^{3} + 26 \, x^{2} e^{2} - 316 \, x e + 312\right )} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}\right )}}{4096 \, {\left (x^{5} e^{6} + 10 \, x^{4} e^{5} + 40 \, x^{3} e^{4} + 80 \, x^{2} e^{3} + 80 \, x e^{2} + 32 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 1.69, size = 109, normalized size = 0.76 \begin {gather*} -\frac {3}{4096} \, \sqrt {3} {\left (\frac {4 \, {\left (3 \, {\left (x e - 2\right )}^{3} \sqrt {-x e + 2} + 44 \, {\left (x e - 2\right )}^{2} \sqrt {-x e + 2} + 176 \, {\left (-x e + 2\right )}^{\frac {3}{2}} - 192 \, \sqrt {-x e + 2}\right )}}{{\left (x e + 2\right )}^{4}} + 3 \, \log \left (\sqrt {-x e + 2} + 2\right ) - 3 \, \log \left (-\sqrt {-x e + 2} + 2\right )\right )} e^{\left (-1\right )} \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 )}^{13/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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