Integrand size = 26, antiderivative size = 94 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {317 \sqrt {1-2 x} \sqrt {3+5 x}}{2200}-\frac {9}{100} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {317 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}} \]
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Time = 0.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {91, 81, 52, 56, 222} \[ \int \frac {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=\frac {317 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}}-\frac {9}{100} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {2 (1-2 x)^{3/2}}{275 \sqrt {5 x+3}}+\frac {317 \sqrt {5 x+3} \sqrt {1-2 x}}{2200} \]
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Rule 52
Rule 56
Rule 81
Rule 91
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {2}{275} \int \frac {\sqrt {1-2 x} \left (\frac {359}{2}+\frac {495 x}{2}\right )}{\sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}-\frac {9}{100} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {317}{440} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {317 \sqrt {1-2 x} \sqrt {3+5 x}}{2200}-\frac {9}{100} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {317}{400} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {317 \sqrt {1-2 x} \sqrt {3+5 x}}{2200}-\frac {9}{100} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {317 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{200 \sqrt {5}} \\ & = -\frac {2 (1-2 x)^{3/2}}{275 \sqrt {3+5 x}}+\frac {317 \sqrt {1-2 x} \sqrt {3+5 x}}{2200}-\frac {9}{100} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {317 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=\frac {10 \sqrt {1-2 x} \left (31+165 x+180 x^2\right )-317 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{2000 \sqrt {3+5 x}} \]
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Time = 3.68 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {\left (1585 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +3600 x^{2} \sqrt {-10 x^{2}-x +3}+951 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3300 x \sqrt {-10 x^{2}-x +3}+620 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{4000 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(99\) |
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Time = 0.24 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=-\frac {317 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (180 \, x^{2} + 165 \, x + 31\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4000 \, {\left (5 \, x + 3\right )}} \]
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\[ \int \frac {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=\int \frac {\sqrt {1 - 2 x} \left (3 x + 2\right )^{2}}{\left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=\frac {317}{4000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {9}{50} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {57}{1000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{125 \, {\left (5 \, x + 3\right )}} \]
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Time = 0.36 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=\frac {3}{5000} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 17 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {317}{2000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{1250 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^2}{{\left (5\,x+3\right )}^{3/2}} \,d x \]
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