Integrand size = 26, antiderivative size = 120 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {239 \sqrt {1-2 x} \sqrt {3+5 x}}{1764 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{42 (2+3 x)^2}+\frac {25}{27} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {17687 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5292 \sqrt {7}} \]
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Time = 0.03 (sec) , antiderivative size = 120, 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, 154, 163, 56, 222, 95, 210} \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {25}{27} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {17687 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{5292 \sqrt {7}}+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{42 (3 x+2)^2}+\frac {239 \sqrt {1-2 x} \sqrt {5 x+3}}{1764 (3 x+2)} \]
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Rule 56
Rule 95
Rule 100
Rule 154
Rule 163
Rule 210
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{42 (2+3 x)^2}-\frac {1}{42} \int \frac {\left (-\frac {387}{2}-350 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx \\ & = \frac {239 \sqrt {1-2 x} \sqrt {3+5 x}}{1764 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{42 (2+3 x)^2}-\frac {1}{882} \int \frac {-\frac {26771}{4}-12250 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx \\ & = \frac {239 \sqrt {1-2 x} \sqrt {3+5 x}}{1764 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{42 (2+3 x)^2}-\frac {17687 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{10584}+\frac {125}{27} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = \frac {239 \sqrt {1-2 x} \sqrt {3+5 x}}{1764 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{42 (2+3 x)^2}-\frac {17687 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{5292}+\frac {1}{27} \left (50 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right ) \\ & = \frac {239 \sqrt {1-2 x} \sqrt {3+5 x}}{1764 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{42 (2+3 x)^2}+\frac {25}{27} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {17687 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5292 \sqrt {7}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.82 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (604+927 x)}{(2+3 x)^2}-34300 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+17687 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{37044} \]
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Time = 1.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.10
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (604+927 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1764 \left (2+3 x \right )^{2} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {25 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{54}-\frac {17687 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{74088}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(132\) |
default | \(-\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (159183 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-308700 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+212244 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -411600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +70748 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-137200 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-38934 x \sqrt {-10 x^{2}-x +3}-25368 \sqrt {-10 x^{2}-x +3}\right )}{74088 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{2}}\) | \(191\) |
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Time = 0.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.13 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {17687 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 34300 \, \sqrt {10} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (927 \, x + 604\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{74088 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
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\[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\int \frac {\left (5 x + 3\right )^{\frac {5}{2}}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{3}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.72 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\frac {25}{54} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {17687}{74088} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{126 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {103 \, \sqrt {-10 \, x^{2} - x + 3}}{588 \, {\left (3 \, x + 2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (90) = 180\).
Time = 0.45 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.66 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=-\frac {17687}{740880} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {25}{54} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {11 \, \sqrt {10} {\left (239 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {85400 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {341600 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{882 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
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Timed out. \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^3} \,d x \]
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