Optimal. Leaf size=57 \[ \frac {x}{54 \sqrt {6} \sqrt {1-2 x} \sqrt {2 x+1}}+\frac {x}{108 \sqrt {6} (1-2 x)^{3/2} (2 x+1)^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {40, 39} \begin {gather*} \frac {x}{54 \sqrt {6} \sqrt {1-2 x} \sqrt {2 x+1}}+\frac {x}{108 \sqrt {6} (1-2 x)^{3/2} (2 x+1)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 39
Rule 40
Rubi steps
\begin {align*} \int \frac {1}{(3-6 x)^{5/2} (2+4 x)^{5/2}} \, dx &=\frac {x}{108 \sqrt {6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac {1}{9} \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx\\ &=\frac {x}{108 \sqrt {6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac {x}{54 \sqrt {6} \sqrt {1-2 x} \sqrt {1+2 x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 0.65 \begin {gather*} \frac {x \left (8 x^2-3\right )}{108 \sqrt {6-12 x} (2 x-1) (2 x+1)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.85, size = 334, normalized size = 5.86 \begin {gather*} \frac {\left (\sqrt {2} \sqrt {2 x+1}-2\right )^9 \left (\frac {91 \left (4 x^2-4 x+1\right )}{294912 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^4}+\frac {35 \left (8 x^3-12 x^2+6 x-1\right )}{36864 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^6}+\frac {91 \left (16 x^4-32 x^3+24 x^2-8 x+1\right )}{73728 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^8}+\frac {5 \left (32 x^5-80 x^4+80 x^3-40 x^2+10 x-1\right )}{18432 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^{10}}-\frac {64 x^6-192 x^5+240 x^4-160 x^3+60 x^2-12 x+1}{55296 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^{12}}+\frac {5 (2 x-1)}{294912 \sqrt {3} \left (\sqrt {2} \sqrt {2 x+1}-2\right )^2}-\frac {1}{3538944 \sqrt {3}}\right )}{(1-2 x)^{3/2} \left (-2 x+\sqrt {2} \sqrt {2 x+1}-1\right )^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 39, normalized size = 0.68 \begin {gather*} -\frac {{\left (8 \, x^{3} - 3 \, x\right )} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{648 \, {\left (16 \, x^{4} - 8 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.02, size = 128, normalized size = 2.25 \begin {gather*} -\frac {1}{82944} \, \sqrt {6} {\left (\frac {{\left (\sqrt {-4 \, x + 2} - 2\right )}^{3}}{{\left (4 \, x + 2\right )}^{\frac {3}{2}}} + \frac {33 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}}{\sqrt {4 \, x + 2}}\right )} - \frac {{\left (4 \, \sqrt {6} {\left (2 \, x + 1\right )} - 9 \, \sqrt {6}\right )} \sqrt {4 \, x + 2} \sqrt {-4 \, x + 2}}{10368 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {6} {\left (4 \, x + 2\right )}^{\frac {3}{2}} {\left (\frac {33 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{2}}{2 \, x + 1} + 2\right )}}{165888 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.61 \begin {gather*} \frac {\left (2 x -1\right ) \left (2 x +1\right ) \left (8 x^{2}-3\right ) x}{3 \left (-6 x +3\right )^{\frac {5}{2}} \left (4 x +2\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 25, normalized size = 0.44 \begin {gather*} \frac {x}{54 \, \sqrt {-24 \, x^{2} + 6}} + \frac {x}{18 \, {\left (-24 \, x^{2} + 6\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 49, normalized size = 0.86 \begin {gather*} -\frac {3\,x\,\sqrt {3-6\,x}-8\,x^3\,\sqrt {3-6\,x}}{\sqrt {4\,x+2}\,\left (-2592\,x^3+1296\,x^2+648\,x-324\right )} \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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