Integrand size = 28, antiderivative size = 249 \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=-\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8120161139 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}}-\frac {486785077 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{248771250 \sqrt {33}} \]
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Time = 0.08 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {103, 159, 164, 114, 120} \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=-\frac {486785077 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{248771250 \sqrt {33}}-\frac {8120161139 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}+\frac {326 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}}{10725}+\frac {2314 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{111375}-\frac {121031 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{30405375}-\frac {3872003 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{30405375}-\frac {486785077 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{547296750} \]
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Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {2}{65} \int \frac {\left (-\frac {111}{2}-\frac {163 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx \\ & = \frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {4 \int \frac {\left (-\frac {5653}{2}-\frac {15041 x}{4}\right ) \sqrt {1-2 x} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx}{10725} \\ & = \frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8 \int \frac {\left (-\frac {518563}{8}-\frac {121031 x}{8}\right ) (3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1447875} \\ & = -\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {8 \int \frac {(3+5 x)^{3/2} \left (\frac {71027395}{16}+\frac {58080045 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{30405375} \\ & = -\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8 \int \frac {\left (-\frac {2382196995}{8}-\frac {7301776155 x}{16}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{456080625} \\ & = -\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {8 \int \frac {\frac {308389708545}{32}+\frac {121802417085 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4104725625} \\ & = -\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {486785077 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{497542500}+\frac {8120161139 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1368241875} \\ & = -\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8120161139 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}}-\frac {486785077 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{248771250 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.45 \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (495379991+2923422930 x-1730459250 x^2-7942630500 x^3+2577015000 x^4+8419950000 x^5\right )+16240322278 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-16727107355 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{8209451250} \]
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Time = 1.36 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.66
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-3788977500000 x^{8}-4064539500000 x^{7}+15771272649 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-16240322278 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+3569208300000 x^{6}+4547296260000 x^{5}-1320576729750 x^{4}-2128036873050 x^{3}-19688021745 x^{2}+315122962755 x +44584199190\right )}{8209451250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(165\) |
risch | \(-\frac {\left (8419950000 x^{5}+2577015000 x^{4}-7942630500 x^{3}-1730459250 x^{2}+2923422930 x +495379991\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{547296750 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {20559313903 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{60202642500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {8120161139 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{15050660625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(267\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {32482477 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6081075}+\frac {495379991 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{547296750}+\frac {20559313903 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{57466158750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {16240322278 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{28733079375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {59161 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{18711}+\frac {2020 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{429}-\frac {168098 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11583}+\frac {200 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{13}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(294\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.30 \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\frac {1}{547296750} \, {\left (8419950000 \, x^{5} + 2577015000 \, x^{4} - 7942630500 \, x^{3} - 1730459250 \, x^{2} + 2923422930 \, x + 495379991\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {551641713241}{738850612500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {8120161139}{4104725625} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
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Timed out. \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\text {Timed out} \]
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\[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \]
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\[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2} \,d x \]
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