Integrand size = 28, antiderivative size = 249 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=-\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac {780320008 \sqrt {1-2 x} \sqrt {3+5 x}}{19253619 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}-\frac {780320008 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}}-\frac {23441272 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1750329 \sqrt {33}} \]
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Time = 0.07 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157, 164, 114, 120} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=-\frac {23441272 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1750329 \sqrt {33}}-\frac {780320008 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{891 (3 x+2)^{9/2}}+\frac {12280 (5 x+3)^{3/2} \sqrt {1-2 x}}{6237 (3 x+2)^{7/2}}+\frac {780320008 \sqrt {5 x+3} \sqrt {1-2 x}}{19253619 \sqrt {3 x+2}}+\frac {11243972 \sqrt {5 x+3} \sqrt {1-2 x}}{2750517 (3 x+2)^{3/2}}-\frac {325796 \sqrt {5 x+3} \sqrt {1-2 x}}{130977 (3 x+2)^{5/2}} \]
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x} \left (-1200+\frac {1005 x}{2}\right )}{(2+3 x)^{9/2}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {8 \int \frac {\left (\frac {232425}{4}-\frac {131115 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx}{18711} \\ & = -\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {16 \int \frac {\frac {7896165}{8}-1154775 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{1964655} \\ & = -\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {32 \int \frac {\frac {347150355}{8}-\frac {210824475 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{41257755} \\ & = -\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac {780320008 \sqrt {1-2 x} \sqrt {3+5 x}}{19253619 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {64 \int \frac {\frac {9262076325}{16}+\frac {7315500075 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{288804285} \\ & = -\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac {780320008 \sqrt {1-2 x} \sqrt {3+5 x}}{19253619 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {11720636 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1750329}+\frac {780320008 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{19253619} \\ & = -\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac {780320008 \sqrt {1-2 x} \sqrt {3+5 x}}{19253619 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}-\frac {780320008 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}}-\frac {23441272 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.72 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.45 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=\frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (13163824553+97637232762 x+289719086787 x^2+429993423180 x^3+319217269302 x^4+94808880972 x^5\right )}{4 (2+3 x)^{11/2}}+i \sqrt {33} \left (97540001 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-100470160 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{57760857} \]
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Time = 1.36 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.31
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {40874 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4546773 \left (\frac {2}{3}+x \right )^{4}}+\frac {181672 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{10609137 \left (\frac {2}{3}+x \right )^{3}}+\frac {11243972 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{24754653 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {7803200080}{19253619} x^{2}-\frac {780320008}{19253619} x +\frac {780320008}{6417873}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {987954808 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{404325999 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1560640016 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{404325999 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {6524 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1948617 \left (\frac {2}{3}+x \right )^{5}}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{649539 \left (\frac {2}{3}+x \right )^{6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(326\) |
default | \(-\frac {2 \left (92076595776 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-94808880972 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+306921985920 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-316029603240 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+409229314560 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-421372804320 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+272819543040 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-280915202880 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+90939847680 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-93638400960 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-2844266429160 x^{7}+12125313024 \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 )-12485120128 \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 )-9860944721976 x^{6}-13004174574558 x^{5}-7108597449432 x^{4}+71666565399 x^{3}+1919645346207 x^{2}+839243621199 x +118474420977\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{57760857 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\) | \(599\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=\frac {2 \, {\left (135 \, {\left (94808880972 \, x^{5} + 319217269302 \, x^{4} + 429993423180 \, x^{3} + 289719086787 \, x^{2} + 97637232762 \, x + 13163824553\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 6627651544 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 17557200180 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{2599238565 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{13/2}} \,d x \]
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