Optimal. Leaf size=85 \[ \frac {x}{405 \sqrt {6} \sqrt {1-2 x} \sqrt {2 x+1}}+\frac {x}{810 \sqrt {6} (1-2 x)^{3/2} (2 x+1)^{3/2}}+\frac {x}{1080 \sqrt {6} (1-2 x)^{5/2} (2 x+1)^{5/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {40, 39} \[ \frac {x}{405 \sqrt {6} \sqrt {1-2 x} \sqrt {2 x+1}}+\frac {x}{810 \sqrt {6} (1-2 x)^{3/2} (2 x+1)^{3/2}}+\frac {x}{1080 \sqrt {6} (1-2 x)^{5/2} (2 x+1)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 39
Rule 40
Rubi steps
\begin {align*} \int \frac {1}{(3-6 x)^{7/2} (2+4 x)^{7/2}} \, dx &=\frac {x}{1080 \sqrt {6} (1-2 x)^{5/2} (1+2 x)^{5/2}}+\frac {2}{15} \int \frac {1}{(3-6 x)^{5/2} (2+4 x)^{5/2}} \, dx\\ &=\frac {x}{1080 \sqrt {6} (1-2 x)^{5/2} (1+2 x)^{5/2}}+\frac {x}{810 \sqrt {6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac {2}{135} \int \frac {1}{(3-6 x)^{3/2} (2+4 x)^{3/2}} \, dx\\ &=\frac {x}{1080 \sqrt {6} (1-2 x)^{5/2} (1+2 x)^{5/2}}+\frac {x}{810 \sqrt {6} (1-2 x)^{3/2} (1+2 x)^{3/2}}+\frac {x}{405 \sqrt {6} \sqrt {1-2 x} \sqrt {1+2 x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.49 \[ \frac {x \left (128 x^4-80 x^2+15\right )}{3240 \sqrt {6-12 x} (1-2 x)^2 (2 x+1)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 49, normalized size = 0.58 \[ -\frac {{\left (128 \, x^{5} - 80 \, x^{3} + 15 \, x\right )} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{19440 \, {\left (64 \, x^{6} - 48 \, x^{4} + 12 \, x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.02, size = 181, normalized size = 2.13 \[ -\frac {1}{39813120} \, \sqrt {6} {\left (\frac {3 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{5}}{{\left (4 \, x + 2\right )}^{\frac {5}{2}}} + \frac {85 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{3}}{{\left (4 \, x + 2\right )}^{\frac {3}{2}}} + \frac {2130 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}}{\sqrt {4 \, x + 2}}\right )} - \frac {{\left ({\left (64 \, \sqrt {6} {\left (2 \, x + 1\right )} - 275 \, \sqrt {6}\right )} {\left (2 \, x + 1\right )} + 300 \, \sqrt {6}\right )} \sqrt {4 \, x + 2} \sqrt {-4 \, x + 2}}{1244160 \, {\left (2 \, x - 1\right )}^{3}} + \frac {\sqrt {6} {\left (\frac {1065 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{4}}{{\left (2 \, x + 1\right )}^{2}} + \frac {85 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{2}}{2 \, x + 1} + 6\right )} {\left (4 \, x + 2\right )}^{\frac {5}{2}}}{79626240 \, {\left (\sqrt {-4 \, x + 2} - 2\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 40, normalized size = 0.47 \[ -\frac {\left (2 x -1\right ) \left (2 x +1\right ) \left (128 x^{4}-80 x^{2}+15\right ) x}{15 \left (-6 x +3\right )^{\frac {7}{2}} \left (4 x +2\right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 37, normalized size = 0.44 \[ \frac {x}{405 \, \sqrt {-24 \, x^{2} + 6}} + \frac {x}{135 \, {\left (-24 \, x^{2} + 6\right )}^{\frac {3}{2}}} + \frac {x}{30 \, {\left (-24 \, x^{2} + 6\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 66, normalized size = 0.78 \[ -\frac {15\,x\,\sqrt {3-6\,x}-80\,x^3\,\sqrt {3-6\,x}+128\,x^5\,\sqrt {3-6\,x}}{\left (\left (6\,x-3\right )\,\left (240\,x+360\right )+1440\right )\,\sqrt {4\,x+2}\,{\left (6\,x-3\right )}^3} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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