Integrand size = 19, antiderivative size = 100 \[ \int (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx=\frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {45}{4} \sqrt {\frac {3}{2}} \arcsin (2 x) \]
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Time = 0.01 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {38, 41, 222} \[ \int (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx=\frac {45}{4} \sqrt {\frac {3}{2}} \arcsin (2 x)+6 \sqrt {6} (1-2 x)^{5/2} x (2 x+1)^{5/2}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {2 x+1} \]
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Rule 38
Rule 41
Rule 222
Rubi steps \begin{align*} \text {integral}& = 6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+5 \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx \\ & = 15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {45}{2} \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx \\ & = \frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {135}{2} \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx \\ & = \frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {135}{2} \int \frac {1}{\sqrt {6-24 x^2}} \, dx \\ & = \frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {45}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x) \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.59 \[ \int (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx=\frac {3}{2} \sqrt {\frac {3}{2}} \left (x \sqrt {1-4 x^2} \left (33-104 x^2+128 x^4\right )+15 \arctan \left (\frac {\sqrt {1-4 x^2}}{1-2 x}\right )\right ) \]
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Time = 0.53 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.07
method | result | size |
risch | \(-\frac {3 x \left (128 x^{4}-104 x^{2}+33\right ) \left (-1+2 x \right ) \left (1+2 x \right ) \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \sqrt {6}}{4 \sqrt {-\left (-1+2 x \right ) \left (1+2 x \right )}\, \sqrt {3-6 x}\, \sqrt {2+4 x}}+\frac {45 \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{8 \sqrt {2+4 x}\, \sqrt {3-6 x}}\) | \(107\) |
default | \(\frac {\left (3-6 x \right )^{\frac {5}{2}} \left (2+4 x \right )^{\frac {7}{2}}}{24}+\frac {\left (3-6 x \right )^{\frac {3}{2}} \left (2+4 x \right )^{\frac {7}{2}}}{8}+\frac {9 \sqrt {3-6 x}\, \left (2+4 x \right )^{\frac {7}{2}}}{32}-\frac {3 \left (2+4 x \right )^{\frac {5}{2}} \sqrt {3-6 x}}{16}-\frac {15 \left (2+4 x \right )^{\frac {3}{2}} \sqrt {3-6 x}}{16}-\frac {45 \sqrt {3-6 x}\, \sqrt {2+4 x}}{8}+\frac {45 \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{8 \sqrt {2+4 x}\, \sqrt {3-6 x}}\) | \(134\) |
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Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.65 \[ \int (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx=\frac {3}{4} \, {\left (128 \, x^{5} - 104 \, x^{3} + 33 \, x\right )} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3} - \frac {45}{8} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {\sqrt {3} \sqrt {2} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{12 \, x}\right ) \]
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Timed out. \[ \int (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.46 \[ \int (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx=\frac {1}{6} \, {\left (-24 \, x^{2} + 6\right )}^{\frac {5}{2}} x + \frac {5}{4} \, {\left (-24 \, x^{2} + 6\right )}^{\frac {3}{2}} x + \frac {45}{4} \, \sqrt {-24 \, x^{2} + 6} x + \frac {45}{8} \, \sqrt {6} \arcsin \left (2 \, x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (70) = 140\).
Time = 0.33 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.27 \[ \int (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx=\frac {3}{40} \, \sqrt {3} \sqrt {2} {\left ({\left ({\left (2 \, {\left ({\left (8 \, {\left (5 \, x - 13\right )} {\left (2 \, x + 1\right )} + 321\right )} {\left (2 \, x + 1\right )} - 451\right )} {\left (2 \, x + 1\right )} + 745\right )} {\left (2 \, x + 1\right )} - 405\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} + 2 \, {\left ({\left (2 \, {\left (3 \, {\left (8 \, x - 17\right )} {\left (2 \, x + 1\right )} + 133\right )} {\left (2 \, x + 1\right )} - 295\right )} {\left (2 \, x + 1\right )} + 195\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} - 20 \, {\left ({\left (4 \, {\left (3 \, x - 5\right )} {\left (2 \, x + 1\right )} + 43\right )} {\left (2 \, x + 1\right )} - 39\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} - 80 \, {\left ({\left (4 \, x - 5\right )} {\left (2 \, x + 1\right )} + 9\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} + 240 \, \sqrt {2 \, x + 1} {\left (x - 1\right )} \sqrt {-2 \, x + 1} + 240 \, \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} + 150 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {2 \, x + 1}\right )\right )} \]
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Timed out. \[ \int (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx=\int {\left (4\,x+2\right )}^{5/2}\,{\left (3-6\,x\right )}^{5/2} \,d x \]
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