Integrand size = 24, antiderivative size = 116 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{\sqrt {3+5 x}} \, dx=\frac {5929 \sqrt {1-2 x} \sqrt {3+5 x}}{16000}+\frac {539 (1-2 x)^{3/2} \sqrt {3+5 x}}{4800}+\frac {49 (1-2 x)^{5/2} \sqrt {3+5 x}}{1200}-\frac {3}{40} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {65219 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{16000 \sqrt {10}} \]
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Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {81, 52, 56, 222} \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{\sqrt {3+5 x}} \, dx=\frac {65219 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{16000 \sqrt {10}}-\frac {3}{40} \sqrt {5 x+3} (1-2 x)^{7/2}+\frac {49 \sqrt {5 x+3} (1-2 x)^{5/2}}{1200}+\frac {539 \sqrt {5 x+3} (1-2 x)^{3/2}}{4800}+\frac {5929 \sqrt {5 x+3} \sqrt {1-2 x}}{16000} \]
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Rule 52
Rule 56
Rule 81
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{40} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {49}{80} \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx \\ & = \frac {49 (1-2 x)^{5/2} \sqrt {3+5 x}}{1200}-\frac {3}{40} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {539}{480} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx \\ & = \frac {539 (1-2 x)^{3/2} \sqrt {3+5 x}}{4800}+\frac {49 (1-2 x)^{5/2} \sqrt {3+5 x}}{1200}-\frac {3}{40} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {5929 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{3200} \\ & = \frac {5929 \sqrt {1-2 x} \sqrt {3+5 x}}{16000}+\frac {539 (1-2 x)^{3/2} \sqrt {3+5 x}}{4800}+\frac {49 (1-2 x)^{5/2} \sqrt {3+5 x}}{1200}-\frac {3}{40} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {65219 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{32000} \\ & = \frac {5929 \sqrt {1-2 x} \sqrt {3+5 x}}{16000}+\frac {539 (1-2 x)^{3/2} \sqrt {3+5 x}}{4800}+\frac {49 (1-2 x)^{5/2} \sqrt {3+5 x}}{1200}-\frac {3}{40} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {65219 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{16000 \sqrt {5}} \\ & = \frac {5929 \sqrt {1-2 x} \sqrt {3+5 x}}{16000}+\frac {539 (1-2 x)^{3/2} \sqrt {3+5 x}}{4800}+\frac {49 (1-2 x)^{5/2} \sqrt {3+5 x}}{1200}-\frac {3}{40} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {65219 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{16000 \sqrt {10}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{\sqrt {3+5 x}} \, dx=\frac {10 \sqrt {1-2 x} \left (64611+116625 x-91180 x^2-90400 x^3+144000 x^4\right )-195657 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{480000 \sqrt {3+5 x}} \]
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Time = 3.87 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {\left (28800 x^{3}-35360 x^{2}+2980 x +21537\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{48000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {65219 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{320000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(103\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (576000 x^{3} \sqrt {-10 x^{2}-x +3}-707200 x^{2} \sqrt {-10 x^{2}-x +3}+195657 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+59600 x \sqrt {-10 x^{2}-x +3}+430740 \sqrt {-10 x^{2}-x +3}\right )}{960000 \sqrt {-10 x^{2}-x +3}}\) | \(104\) |
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Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.62 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{\sqrt {3+5 x}} \, dx=\frac {1}{48000} \, {\left (28800 \, x^{3} - 35360 \, x^{2} + 2980 \, x + 21537\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {65219}{320000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
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\[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{\sqrt {3+5 x}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \cdot \left (3 x + 2\right )}{\sqrt {5 x + 3}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{\sqrt {3+5 x}} \, dx=\frac {3}{5} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {221}{300} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {149}{2400} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {65219}{320000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {7179}{16000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (83) = 166\).
Time = 0.31 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.75 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{\sqrt {3+5 x}} \, dx=\frac {1}{800000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1}{30000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1}{400} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)}{\sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,\left (3\,x+2\right )}{\sqrt {5\,x+3}} \,d x \]
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