Optimal. Leaf size=73 \[ -\frac {3 \sqrt {3} (2-e x)^{3/2}}{e (e x+2)}-\frac {9 \sqrt {3} \sqrt {2-e x}}{e}+\frac {18 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e} \]
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Rubi [A] time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {627, 47, 50, 63, 206} \begin {gather*} -\frac {3 \sqrt {3} (2-e x)^{3/2}}{e (e x+2)}-\frac {9 \sqrt {3} \sqrt {2-e x}}{e}+\frac {18 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
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)^{7/2}} \, dx &=\int \frac {(6-3 e x)^{3/2}}{(2+e x)^2} \, dx\\ &=-\frac {3 \sqrt {3} (2-e x)^{3/2}}{e (2+e x)}-\frac {9}{2} \int \frac {\sqrt {6-3 e x}}{2+e x} \, dx\\ &=-\frac {9 \sqrt {3} \sqrt {2-e x}}{e}-\frac {3 \sqrt {3} (2-e x)^{3/2}}{e (2+e x)}-54 \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx\\ &=-\frac {9 \sqrt {3} \sqrt {2-e x}}{e}-\frac {3 \sqrt {3} (2-e x)^{3/2}}{e (2+e x)}+\frac {36 \operatorname {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{e}\\ &=-\frac {9 \sqrt {3} \sqrt {2-e x}}{e}-\frac {3 \sqrt {3} (2-e x)^{3/2}}{e (2+e x)}+\frac {18 \sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{e}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 55, normalized size = 0.75 \begin {gather*} -\frac {3 (e x-2)^2 \sqrt {12-3 e^2 x^2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {1}{2}-\frac {e x}{4}\right )}{40 e \sqrt {e x+2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.39, size = 177, normalized size = 2.42 \begin {gather*} \frac {\frac {\left (\frac {6 \sqrt {3} (e x+2)}{e}+\frac {12 \sqrt {3}}{e}\right ) \sqrt {4 (e x+2)-(e x+2)^2}}{\sqrt {e x+2}}-\frac {18 \sqrt {3} (e x+2) \tanh ^{-1}\left (\frac {2 \sqrt {e x+2}}{\sqrt {4 (e x+2)-(e x+2)^2}}\right )}{e}}{\left (\frac {\sqrt {4 (e x+2)-(e x+2)^2}}{\sqrt {e x+2}}-2\right ) \left (\frac {\sqrt {4 (e x+2)-(e x+2)^2}}{\sqrt {e x+2}}+2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 122, normalized size = 1.67 \begin {gather*} \frac {3 \, {\left (3 \, \sqrt {3} {\left (e^{2} x^{2} + 4 \, e x + 4\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 ) - 2 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 4\right )} \sqrt {e x + 2}\right )}}{e^{3} x^{2} + 4 \, e^{2} x + 4 \, e} \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 = 0.06, size = 101, normalized size = 1.38 \begin {gather*} \frac {6 \sqrt {-e^{2} x^{2}+4}\, \left (3 \sqrt {3}\, e x \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )-\sqrt {-3 e x +6}\, e x +6 \sqrt {3}\, \arctanh \left (\frac {\sqrt {3}\, \sqrt {-3 e x +6}}{6}\right )-4 \sqrt {-3 e x +6}\right ) \sqrt {3}}{\sqrt {\left (e x +2\right )^{3}}\, \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 {7}{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 )}^{7/2}} \,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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