Integrand size = 26, antiderivative size = 115 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {74}{45} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {346}{135} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {175}{27} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {100, 159, 163, 56, 222, 95, 210} \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {346}{135} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {175}{27} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+\frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)}+\frac {74}{45} \sqrt {5 x+3} \sqrt {1-2 x} \]
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Rule 56
Rule 95
Rule 100
Rule 159
Rule 163
Rule 210
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1}{3} \int \frac {\sqrt {1-2 x} \left (\frac {157}{2}+74 x\right )}{(2+3 x) \sqrt {3+5 x}} \, dx \\ & = \frac {74}{45} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1}{45} \int \frac {\frac {2503}{2}+346 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = \frac {74}{45} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {346}{135} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {1225}{54} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = \frac {74}{45} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1225}{27} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {692 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{135 \sqrt {5}} \\ & = \frac {74}{45} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {346}{135} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {175}{27} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {1}{675} \left (\frac {15 \sqrt {1-2 x} \left (759+1301 x+60 x^2\right )}{(2+3 x) \sqrt {3+5 x}}-346 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-4375 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right ) \]
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Time = 1.19 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.16
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (253+12 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{45 \left (2+3 x \right ) \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {173 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{675}-\frac {175 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{54}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(133\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1038 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +13125 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +692 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+8750 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+360 x \sqrt {-10 x^{2}-x +3}+7590 \sqrt {-10 x^{2}-x +3}\right )}{1350 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )}\) | \(146\) |
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Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=-\frac {346 \, \sqrt {5} \sqrt {2} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 4375 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 30 \, {\left (12 \, x + 253\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1350 \, {\left (3 \, x + 2\right )}} \]
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\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{2} \sqrt {5 x + 3}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {173}{675} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {175}{54} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {4}{45} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{9 \, {\left (3 \, x + 2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (83) = 166\).
Time = 0.39 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.43 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {35}{108} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {173}{675} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {4}{225} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {1078 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^2\,\sqrt {5\,x+3}} \,d x \]
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