Optimal. Leaf size=74 \[ 3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac {9}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {2 x+1}+\frac {9}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x) \]
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Rubi [A] time = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {38, 41, 216} \begin {gather*} 3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac {9}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {2 x+1}+\frac {9}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 216
Rubi steps
\begin {align*} \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx &=3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac {9}{2} \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx\\ &=\frac {9}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac {27}{2} \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx\\ &=\frac {9}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac {27}{2} \int \frac {1}{\sqrt {6-24 x^2}} \, dx\\ &=\frac {9}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac {9}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 39, normalized size = 0.53 \begin {gather*} \frac {3}{4} \sqrt {\frac {3}{2}} \left (2 x \sqrt {1-4 x^2} \left (5-8 x^2\right )+3 \sin ^{-1}(2 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.84, size = 179, normalized size = 2.42 \begin {gather*} \frac {-12 \sqrt {6} \sqrt {1-2 x} x \sqrt {2 x+1} \left (8 x^2-5\right ) \left (-56 x^3-364 x^2-490 x-169\right )-12 \sqrt {3} \sqrt {1-2 x} x \left (8 x^2-5\right ) \left (16 x^4+368 x^3+1088 x^2+932 x+239\right )}{-896 x^3-5824 x^2+\sqrt {2} \sqrt {2 x+1} \left (64 x^3+1440 x^2+3632 x+1912\right )-7840 x-2704}+9 \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {2 x+1}-\sqrt {2}}{\sqrt {1-2 x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 60, normalized size = 0.81 \begin {gather*} -\frac {3}{4} \, {\left (8 \, x^{3} - 5 \, x\right )} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3} - \frac {9}{8} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {\sqrt {3} \sqrt {2} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{12 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.98, size = 125, normalized size = 1.69 \begin {gather*} -\frac {1}{8} \, \sqrt {3} \sqrt {2} {\left ({\left ({\left (4 \, {\left (3 \, x - 5\right )} {\left (2 \, x + 1\right )} + 43\right )} {\left (2 \, x + 1\right )} - 39\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} + 4 \, {\left ({\left (4 \, x - 5\right )} {\left (2 \, x + 1\right )} + 9\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} - 24 \, \sqrt {2 \, x + 1} {\left (x - 1\right )} \sqrt {-2 \, x + 1} - 24 \, \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} - 18 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {2 \, x + 1}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 102, normalized size = 1.38 \begin {gather*} \frac {9 \sqrt {\left (4 x +2\right ) \left (-6 x +3\right )}\, \sqrt {6}\, \arcsin \left (2 x \right )}{8 \sqrt {4 x +2}\, \sqrt {-6 x +3}}+\frac {\left (-6 x +3\right )^{\frac {3}{2}} \left (4 x +2\right )^{\frac {5}{2}}}{16}+\frac {3 \left (4 x +2\right )^{\frac {5}{2}} \sqrt {-6 x +3}}{16}-\frac {3 \left (4 x +2\right )^{\frac {3}{2}} \sqrt {-6 x +3}}{16}-\frac {9 \sqrt {-6 x +3}\, \sqrt {4 x +2}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.86, size = 34, normalized size = 0.46 \begin {gather*} \frac {1}{4} \, {\left (-24 \, x^{2} + 6\right )}^{\frac {3}{2}} x + \frac {9}{4} \, \sqrt {-24 \, x^{2} + 6} x + \frac {9}{8} \, \sqrt {6} \arcsin \left (2 \, x\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 {\left (4\,x+2\right )}^{3/2}\,{\left (3-6\,x\right )}^{3/2} \,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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