Integrand size = 26, antiderivative size = 165 \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {9568559 \sqrt {1-2 x} \sqrt {3+5 x}}{12800000}+\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {105254149 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800000 \sqrt {10}} \]
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Time = 0.04 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {92, 81, 52, 56, 222} \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {105254149 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800000 \sqrt {10}}-\frac {1}{20} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {193 (5 x+3)^{3/2} (1-2 x)^{7/2}}{2000}-\frac {7189 \sqrt {5 x+3} (1-2 x)^{7/2}}{32000}+\frac {79079 \sqrt {5 x+3} (1-2 x)^{5/2}}{960000}+\frac {869869 \sqrt {5 x+3} (1-2 x)^{3/2}}{3840000}+\frac {9568559 \sqrt {5 x+3} \sqrt {1-2 x}}{12800000} \]
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Rule 52
Rule 56
Rule 81
Rule 92
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}-\frac {1}{60} \int \left (-186-\frac {579 x}{2}\right ) (1-2 x)^{5/2} \sqrt {3+5 x} \, dx \\ & = -\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {7189 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{4000} \\ & = -\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {79079 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{64000} \\ & = \frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {869869 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{384000} \\ & = \frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {9568559 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{2560000} \\ & = \frac {9568559 \sqrt {1-2 x} \sqrt {3+5 x}}{12800000}+\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {105254149 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{25600000} \\ & = \frac {9568559 \sqrt {1-2 x} \sqrt {3+5 x}}{12800000}+\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {105254149 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{12800000 \sqrt {5}} \\ & = \frac {9568559 \sqrt {1-2 x} \sqrt {3+5 x}}{12800000}+\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {105254149 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800000 \sqrt {10}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.53 \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {10 \sqrt {1-2 x} \left (27911781+354090375 x+328830220 x^2-1017874400 x^3-902544000 x^4+1163520000 x^5+1152000000 x^6\right )-315762447 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{384000000 \sqrt {3+5 x}} \]
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Time = 1.15 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {\left (230400000 x^{5}+94464000 x^{4}-237187200 x^{3}-61262560 x^{2}+102523580 x +9303927\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{38400000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {105254149 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{256000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(113\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (4608000000 x^{5} \sqrt {-10 x^{2}-x +3}+1889280000 x^{4} \sqrt {-10 x^{2}-x +3}-4743744000 x^{3} \sqrt {-10 x^{2}-x +3}-1225251200 x^{2} \sqrt {-10 x^{2}-x +3}+315762447 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2050471600 x \sqrt {-10 x^{2}-x +3}+186078540 \sqrt {-10 x^{2}-x +3}\right )}{768000000 \sqrt {-10 x^{2}-x +3}}\) | \(138\) |
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Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.50 \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {1}{38400000} \, {\left (230400000 \, x^{5} + 94464000 \, x^{4} - 237187200 \, x^{3} - 61262560 \, x^{2} + 102523580 \, x + 9303927\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {105254149}{256000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\int \left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{2} \sqrt {5 x + 3}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.63 \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=-\frac {3}{5} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {93}{500} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {18251}{40000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {27893}{480000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {869869}{640000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {105254149}{256000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {869869}{12800000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (120) = 240\).
Time = 0.36 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.16 \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\frac {3}{640000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {7}{40000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {79}{9600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {89}{120000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{250} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {6}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
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Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2\,\sqrt {5\,x+3} \,d x \]
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