Integrand size = 28, antiderivative size = 218 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=-\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {146222113 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3827250 \sqrt {33}}-\frac {1654421 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1913625 \sqrt {33}} \]
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Time = 0.06 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {103, 159, 164, 114, 120} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=-\frac {1654421 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1913625 \sqrt {33}}-\frac {146222113 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3827250 \sqrt {33}}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {9698 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{93555}-\frac {146963 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{467775}-\frac {1654421 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{4209975} \]
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Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2}{33} \int \frac {\left (-50-\frac {185 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx \\ & = \frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-\frac {9245}{4}-\frac {24245 x}{4}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx}{4455} \\ & = \frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (\frac {84395}{4}-\frac {2204445 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{467775} \\ & = -\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\sqrt {3+5 x} \left (\frac {44328915}{16}+\frac {24816315 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{7016625} \\ & = -\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {-\frac {685297455}{8}-\frac {2193331695 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{63149625} \\ & = -\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {1654421 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3827250}+\frac {146222113 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{42099750} \\ & = -\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {146222113 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3827250 \sqrt {33}}-\frac {1654421 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1913625 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.50 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (3748468+9143865 x-16381350 x^2-12379500 x^3+25515000 x^4\right )+146222113 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-149530955 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{126299250} \]
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Time = 1.36 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.73
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-22963500000 x^{7}+140986329 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-146222113 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-6463800000 x^{6}+28643220000 x^{5}+5066658000 x^{4}-15351281550 x^{3}-3614874270 x^{2}+2433073980 x +674724240\right )}{126299250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(160\) |
risch | \(-\frac {\left (25515000 x^{4}-12379500 x^{3}-16381350 x^{2}+9143865 x +3748468\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{4209975 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {45686497 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{231548625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {146222113 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{463097250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(262\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {203197 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{93555}+\frac {3748468 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4209975}+\frac {91372994 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{442047375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {146222113 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{442047375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {72806 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{18711}+\frac {200 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{33}-\frac {2620 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{891}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(272\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.32 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=\frac {1}{4209975} \, {\left (25515000 \, x^{4} - 12379500 \, x^{3} - 16381350 \, x^{2} + 9143865 \, x + 3748468\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {4860460861}{11366932500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {146222113}{126299250} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2}} \,d x } \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {3\,x+2}} \,d x \]
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