Integrand size = 26, antiderivative size = 86 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{3/2}} \, dx=-\frac {22 \sqrt {1-2 x}}{5 \sqrt {3+5 x}}+\frac {4}{15} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {14}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 163, 56, 222, 95, 210} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{3/2}} \, dx=\frac {4}{15} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {14}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {22 \sqrt {1-2 x}}{5 \sqrt {5 x+3}} \]
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Rule 56
Rule 95
Rule 100
Rule 163
Rule 210
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {22 \sqrt {1-2 x}}{5 \sqrt {3+5 x}}-\frac {2}{5} \int \frac {\frac {79}{2}-2 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {22 \sqrt {1-2 x}}{5 \sqrt {3+5 x}}+\frac {4}{15} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx-\frac {49}{3} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = -\frac {22 \sqrt {1-2 x}}{5 \sqrt {3+5 x}}-\frac {98}{3} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {8 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{15 \sqrt {5}} \\ & = -\frac {22 \sqrt {1-2 x}}{5 \sqrt {3+5 x}}+\frac {4}{15} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {14}{3} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \\ \end{align*}
Time = 1.20 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.86 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{3/2}} \, dx=\frac {2}{75} \left (-\frac {165 \sqrt {1-2 x}}{\sqrt {3+5 x}}-175 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )-4 \sqrt {10} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )-175 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(123\) vs. \(2(60)=120\).
Time = 1.15 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.44
method | result | size |
default | \(\frac {\left (10 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -875 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +6 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-525 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-330 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{75 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(124\) |
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Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (60) = 120\).
Time = 0.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.42 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{3/2}} \, dx=-\frac {2 \, \sqrt {5} \sqrt {2} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 175 \, \sqrt {7} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 330 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{75 \, {\left (5 \, x + 3\right )}} \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{3/2}} \, dx=\frac {2}{75} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {7}{3} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {44 \, x}{5 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {22}{5 \, \sqrt {-10 \, x^{2} - x + 3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (60) = 120\).
Time = 0.43 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.33 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{3/2}} \, dx=-\frac {7}{30} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {2}{75} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11}{50} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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