Integrand size = 24, antiderivative size = 74 \[ \int \frac {\sqrt {1-2 x} (2+3 x)}{(3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{3/2}}{165 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x}}{25 \sqrt {3+5 x}}-\frac {6}{25} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {79, 49, 56, 222} \[ \int \frac {\sqrt {1-2 x} (2+3 x)}{(3+5 x)^{5/2}} \, dx=-\frac {6}{25} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2}}{165 (5 x+3)^{3/2}}-\frac {6 \sqrt {1-2 x}}{25 \sqrt {5 x+3}} \]
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Rule 49
Rule 56
Rule 79
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{3/2}}{165 (3+5 x)^{3/2}}+\frac {3}{5} \int \frac {\sqrt {1-2 x}}{(3+5 x)^{3/2}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2}}{165 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x}}{25 \sqrt {3+5 x}}-\frac {6}{25} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2}}{165 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x}}{25 \sqrt {3+5 x}}-\frac {12 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{25 \sqrt {5}} \\ & = -\frac {2 (1-2 x)^{3/2}}{165 (3+5 x)^{3/2}}-\frac {6 \sqrt {1-2 x}}{25 \sqrt {3+5 x}}-\frac {6}{25} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {1-2 x} (2+3 x)}{(3+5 x)^{5/2}} \, dx=-\frac {2 \sqrt {1-2 x} (302+485 x)}{825 (3+5 x)^{3/2}}+\frac {6}{25} \sqrt {\frac {2}{5}} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right ) \]
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Time = 1.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.30
method | result | size |
default | \(-\frac {\left (2475 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+2970 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +891 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+4850 x \sqrt {-10 x^{2}-x +3}+3020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{4125 \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\) | \(96\) |
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Time = 0.23 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {1-2 x} (2+3 x)}{(3+5 x)^{5/2}} \, dx=\frac {99 \, \sqrt {5} \sqrt {2} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 10 \, {\left (485 \, x + 302\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
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\[ \int \frac {\sqrt {1-2 x} (2+3 x)}{(3+5 x)^{5/2}} \, dx=\int \frac {\sqrt {1 - 2 x} \left (3 x + 2\right )}{\left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.33 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {1-2 x} (2+3 x)}{(3+5 x)^{5/2}} \, dx=-\frac {4 \, \sqrt {-10 \, x^{2} - x + 3}}{15 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {8 \, \sqrt {-10 \, x^{2} - x + 3}}{165 \, {\left (5 \, x + 3\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (51) = 102\).
Time = 0.38 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.89 \[ \int \frac {\sqrt {1-2 x} (2+3 x)}{(3+5 x)^{5/2}} \, dx=-\frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{66000 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}}} - \frac {6}{125} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {13 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{1100 \, \sqrt {5 \, x + 3}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {195 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{4125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (2+3 x)}{(3+5 x)^{5/2}} \, dx=\int \frac {\sqrt {1-2\,x}\,\left (3\,x+2\right )}{{\left (5\,x+3\right )}^{5/2}} \,d x \]
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