Integrand size = 28, antiderivative size = 280 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=-\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {3+5 x}}{750891141 (2+3 x)^{3/2}}+\frac {129922578224 \sqrt {1-2 x} \sqrt {3+5 x}}{5256237987 \sqrt {2+3 x}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}-\frac {129922578224 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{477839817 \sqrt {33}}-\frac {3894280616 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{477839817 \sqrt {33}} \]
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Time = 0.08 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157, 164, 114, 120} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=-\frac {3894280616 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{477839817 \sqrt {33}}-\frac {129922578224 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{477839817 \sqrt {33}}+\frac {60080 \sqrt {1-2 x} (5 x+3)^{5/2}}{34749 (3 x+2)^{9/2}}+\frac {370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1287 (3 x+2)^{11/2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}-\frac {2622980 \sqrt {1-2 x} (5 x+3)^{3/2}}{1702701 (3 x+2)^{7/2}}+\frac {129922578224 \sqrt {1-2 x} \sqrt {5 x+3}}{5256237987 \sqrt {3 x+2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {5 x+3}}{750891141 (3 x+2)^{3/2}}-\frac {54281308 \sqrt {1-2 x} \sqrt {5 x+3}}{35756721 (3 x+2)^{5/2}} \]
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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)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {2}{39} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}-\frac {4 \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2} \left (-1945+\frac {1675 x}{2}\right )}{(2+3 x)^{11/2}} \, dx}{1287} \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac {8 \int \frac {\left (\frac {375445}{4}-\frac {210225 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx}{34749} \\ & = -\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac {16 \int \frac {\left (\frac {38522835}{8}-5499150 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx}{5108103} \\ & = -\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac {32 \int \frac {\frac {1283806245}{16}-\frac {796081425 x}{8}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{536350815} \\ & = -\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {3+5 x}}{750891141 (2+3 x)^{3/2}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac {64 \int \frac {\frac {14437282485}{4}-\frac {35178722175 x}{16}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{11263367115} \\ & = -\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {3+5 x}}{750891141 (2+3 x)^{3/2}}+\frac {129922578224 \sqrt {1-2 x} \sqrt {3+5 x}}{5256237987 \sqrt {2+3 x}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac {128 \int \frac {\frac {1541948542725}{32}+\frac {609012085425 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{78843569805} \\ & = -\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {3+5 x}}{750891141 (2+3 x)^{3/2}}+\frac {129922578224 \sqrt {1-2 x} \sqrt {3+5 x}}{5256237987 \sqrt {2+3 x}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac {1947140308 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{477839817}+\frac {129922578224 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5256237987} \\ & = -\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {3+5 x}}{750891141 (2+3 x)^{3/2}}+\frac {129922578224 \sqrt {1-2 x} \sqrt {3+5 x}}{5256237987 \sqrt {2+3 x}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}-\frac {129922578224 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{477839817 \sqrt {33}}-\frac {3894280616 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{477839817 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.94 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.41 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (4382625184685+39086872650957 x+145238558453649 x^2+287874442427697 x^3+321056742490902 x^4+191022825888450 x^5+47356779762648 x^6\right )}{4 (2+3 x)^{13/2}}+i \sqrt {33} \left (16240322278 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-16727107355 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{15768713961} \]
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Time = 1.31 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.25
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6908733 \left (\frac {2}{3}+x \right )^{7}}+\frac {518 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{938223 \left (\frac {2}{3}+x \right )^{6}}-\frac {478462 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{75996063 \left (\frac {2}{3}+x \right )^{5}}+\frac {17427370 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1241269029 \left (\frac {2}{3}+x \right )^{4}}+\frac {13028276 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2896294401 \left (\frac {2}{3}+x \right )^{3}}+\frac {1876198516 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6758020269 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {1299225782240}{5256237987} x^{2}-\frac {129922578224}{5256237987} x +\frac {129922578224}{1752079329}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {164474511224 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{110380997727 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {259845156448 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{110380997727 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(350\) |
default | \(\frac {2 \left (47356779762648 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-45989031044484 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+189427119050592 \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}-183956124177936 \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}+315711865084320 \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}-306593540296560 \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}+280632768963840 \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}-272527591374720 \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}+140316384481920 \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}-136263795687360 \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}+1420703392879440 x^{8}+37417702528512 \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}-36337012183296 \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}+5872755115941444 x^{7}+4157522503168 \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 )-4037445798144 \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 )+9778559734528578 x^{6}+7880198067307566 x^{5}+2331269398474443 x^{4}-982548126959616 x^{3}-1058407652589420 x^{2}-338633978304558 x -39443626662165\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{15768713961 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {13}{2}}}\) | \(694\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\frac {2 \, {\left (135 \, {\left (47356779762648 \, x^{6} + 191022825888450 \, x^{5} + 321056742490902 \, x^{4} + 287874442427697 \, x^{3} + 145238558453649 \, x^{2} + 39086872650957 \, x + 4382625184685\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1103283426482 \, \sqrt {-30} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 2923258010040 \, \sqrt {-30} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{709592128245 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {15}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {15}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{15/2}} \,d x \]
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