Integrand size = 26, antiderivative size = 173 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=-\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1403963 \sqrt {1-2 x}}{3136 (2+3 x) \sqrt {3+5 x}}+\frac {145708761 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}} \]
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Time = 0.05 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 156, 157, 12, 95, 210} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\frac {145708761 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3136 \sqrt {7}}-\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {5 x+3}}+\frac {1403963 \sqrt {1-2 x}}{3136 (3 x+2) \sqrt {5 x+3}}+\frac {8063 \sqrt {1-2 x}}{224 (3 x+2)^2 \sqrt {5 x+3}}+\frac {33 \sqrt {1-2 x}}{8 (3 x+2)^3 \sqrt {5 x+3}}+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}} \]
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 157
Rule 210
Rubi steps \begin{align*} \text {integral}& = \frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {1}{12} \int \frac {\frac {341}{2}-264 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx \\ & = \frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {1}{252} \int \frac {\frac {86163}{4}-31185 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx \\ & = \frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {\int \frac {\frac {15937383}{8}-2539845 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx}{3528} \\ & = \frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1403963 \sqrt {1-2 x}}{3136 (2+3 x) \sqrt {3+5 x}}+\frac {\int \frac {\frac {1880555061}{16}-\frac {442248345 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{24696} \\ & = -\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1403963 \sqrt {1-2 x}}{3136 (2+3 x) \sqrt {3+5 x}}-\frac {\int \frac {100976171373}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{135828} \\ & = -\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1403963 \sqrt {1-2 x}}{3136 (2+3 x) \sqrt {3+5 x}}-\frac {145708761 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6272} \\ & = -\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1403963 \sqrt {1-2 x}}{3136 (2+3 x) \sqrt {3+5 x}}-\frac {145708761 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{3136} \\ & = -\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1403963 \sqrt {1-2 x}}{3136 (2+3 x) \sqrt {3+5 x}}+\frac {145708761 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}} \\ \end{align*}
Time = 3.04 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\frac {-\frac {7 \sqrt {1-2 x} \left (327908240+1985778980 x+4508028900 x^2+4546951839 x^3+1719322065 x^4\right )}{(2+3 x)^4 \sqrt {3+5 x}}-145708761 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )-145708761 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )}{21952} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(134)=268\).
Time = 1.14 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.72
method | result | size |
default | \(-\frac {\left (59012048205 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+192772690803 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+251784739008 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+24070508910 x^{4} \sqrt {-10 x^{2}-x +3}+164359482408 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+63657325746 x^{3} \sqrt {-10 x^{2}-x +3}+53620824048 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +63112404600 x^{2} \sqrt {-10 x^{2}-x +3}+6994020528 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+27800905720 x \sqrt {-10 x^{2}-x +3}+4590715360 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{43904 \left (2+3 x \right )^{4} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\) | \(298\) |
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Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\frac {145708761 \, \sqrt {7} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1719322065 \, x^{4} + 4546951839 \, x^{3} + 4508028900 \, x^{2} + 1985778980 \, x + 327908240\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43904 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{5} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (134) = 268\).
Time = 0.30 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.71 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=-\frac {145708761}{43904} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {63678595 \, x}{4704 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {66486521}{9408 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {49}{36 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {665}{72 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {7799}{96 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {457237}{448 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (134) = 268\).
Time = 0.55 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.47 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=-\frac {145708761}{439040} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {275}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {11 \, \sqrt {10} {\left (13252949 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 8830442040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 2086818820800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {170309125952000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {681236503808000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1568 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^5\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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