Optimal. Leaf size=85 \[ \frac {2 (2-e x)^{7/2}}{7 \sqrt {3} e}-\frac {8 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {32 (2-e x)^{3/2}}{\sqrt {3} e}-\frac {128 \sqrt {2-e x}}{\sqrt {3} e} \]
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Rubi [A] time = 0.02, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {627, 43} \begin {gather*} \frac {2 (2-e x)^{7/2}}{7 \sqrt {3} e}-\frac {8 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {32 (2-e x)^{3/2}}{\sqrt {3} e}-\frac {128 \sqrt {2-e x}}{\sqrt {3} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int \frac {(2+e x)^{7/2}}{\sqrt {12-3 e^2 x^2}} \, dx &=\int \frac {(2+e x)^3}{\sqrt {6-3 e x}} \, dx\\ &=\int \left (\frac {64}{\sqrt {6-3 e x}}-16 \sqrt {6-3 e x}+\frac {4}{3} (6-3 e x)^{3/2}-\frac {1}{27} (6-3 e x)^{5/2}\right ) \, dx\\ &=-\frac {128 \sqrt {2-e x}}{\sqrt {3} e}+\frac {32 (2-e x)^{3/2}}{\sqrt {3} e}-\frac {8 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {2 (2-e x)^{7/2}}{7 \sqrt {3} e}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 57, normalized size = 0.67 \begin {gather*} \frac {2 (e x-2) \sqrt {e x+2} \left (5 e^3 x^3+54 e^2 x^2+284 e x+1416\right )}{35 e \sqrt {12-3 e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 85, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {4 (e x+2)-(e x+2)^2} \left (5 \sqrt {3} (e x+2)^3+24 \sqrt {3} (e x+2)^2+128 \sqrt {3} (e x+2)+1024 \sqrt {3}\right )}{105 e \sqrt {e x+2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 54, normalized size = 0.64 \begin {gather*} -\frac {2 \, {\left (5 \, e^{3} x^{3} + 54 \, e^{2} x^{2} + 284 \, e x + 1416\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{105 \, {\left (e^{2} x + 2 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + 2\right )}^{\frac {7}{2}}}{\sqrt {-3 \, e^{2} x^{2} + 12}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 52, normalized size = 0.61 \begin {gather*} \frac {2 \left (e x -2\right ) \left (5 e^{3} x^{3}+54 e^{2} x^{2}+284 e x +1416\right ) \sqrt {e x +2}}{35 \sqrt {-3 e^{2} x^{2}+12}\, e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 3.18, size = 45, normalized size = 0.53 \begin {gather*} -\frac {2 i \, \sqrt {3} {\left (5 \, e^{4} x^{4} + 44 \, e^{3} x^{3} + 176 \, e^{2} x^{2} + 848 \, e x - 2832\right )}}{105 \, \sqrt {e x - 2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 74, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {944\,\sqrt {e\,x+2}}{35\,e^2}+\frac {36\,x^2\,\sqrt {e\,x+2}}{35}+\frac {568\,x\,\sqrt {e\,x+2}}{105\,e}+\frac {2\,e\,x^3\,\sqrt {e\,x+2}}{21}\right )}{x+\frac {2}{e}} \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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