Integrand size = 28, antiderivative size = 311 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=-\frac {112817764 \sqrt {1-2 x} \sqrt {3+5 x}}{107270163 (2+3 x)^{7/2}}+\frac {3914701972 \sqrt {1-2 x} \sqrt {3+5 x}}{3754455705 (2+3 x)^{5/2}}+\frac {181941877952 \sqrt {1-2 x} \sqrt {3+5 x}}{26281189935 (2+3 x)^{3/2}}+\frac {12641611554328 \sqrt {1-2 x} \sqrt {3+5 x}}{183968329545 \sqrt {2+3 x}}-\frac {1085156 \sqrt {1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac {16636 \sqrt {1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}-\frac {12641611554328 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16724393595 \sqrt {33}}-\frac {380220959152 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{16724393595 \sqrt {33}} \]
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Time = 0.10 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157, 164, 114, 120} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=-\frac {380220959152 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{16724393595 \sqrt {33}}-\frac {12641611554328 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16724393595 \sqrt {33}}+\frac {16636 \sqrt {1-2 x} (5 x+3)^{5/2}}{11583 (3 x+2)^{11/2}}+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{351 (3 x+2)^{13/2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}-\frac {1085156 \sqrt {1-2 x} (5 x+3)^{3/2}}{729729 (3 x+2)^{9/2}}+\frac {12641611554328 \sqrt {1-2 x} \sqrt {5 x+3}}{183968329545 \sqrt {3 x+2}}+\frac {181941877952 \sqrt {1-2 x} \sqrt {5 x+3}}{26281189935 (3 x+2)^{3/2}}+\frac {3914701972 \sqrt {1-2 x} \sqrt {5 x+3}}{3754455705 (3 x+2)^{5/2}}-\frac {112817764 \sqrt {1-2 x} \sqrt {5 x+3}}{107270163 (3 x+2)^{7/2}} \]
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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}}{45 (2+3 x)^{15/2}}+\frac {2}{45} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}-\frac {4 \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2} \left (-\frac {4715}{2}+\frac {3325 x}{2}\right )}{(2+3 x)^{13/2}} \, dx}{1755} \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac {16636 \sqrt {1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac {8 \int \frac {\left (\frac {712045}{4}-241650 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx}{57915} \\ & = -\frac {1085156 \sqrt {1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac {16636 \sqrt {1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac {16 \int \frac {\left (\frac {73680705}{8}-\frac {50506125 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx}{10945935} \\ & = -\frac {112817764 \sqrt {1-2 x} \sqrt {3+5 x}}{107270163 (2+3 x)^{7/2}}-\frac {1085156 \sqrt {1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac {16636 \sqrt {1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac {32 \int \frac {\frac {2496930465}{16}-\frac {898667625 x}{4}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{1609052445} \\ & = -\frac {112817764 \sqrt {1-2 x} \sqrt {3+5 x}}{107270163 (2+3 x)^{7/2}}+\frac {3914701972 \sqrt {1-2 x} \sqrt {3+5 x}}{3754455705 (2+3 x)^{5/2}}-\frac {1085156 \sqrt {1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac {16636 \sqrt {1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac {64 \int \frac {\frac {97169848605}{8}-\frac {220201985925 x}{16}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{56316835575} \\ & = -\frac {112817764 \sqrt {1-2 x} \sqrt {3+5 x}}{107270163 (2+3 x)^{7/2}}+\frac {3914701972 \sqrt {1-2 x} \sqrt {3+5 x}}{3754455705 (2+3 x)^{5/2}}+\frac {181941877952 \sqrt {1-2 x} \sqrt {3+5 x}}{26281189935 (2+3 x)^{3/2}}-\frac {1085156 \sqrt {1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac {16636 \sqrt {1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac {128 \int \frac {\frac {16880201241165}{32}-\frac {639639414675 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{1182653547075} \\ & = -\frac {112817764 \sqrt {1-2 x} \sqrt {3+5 x}}{107270163 (2+3 x)^{7/2}}+\frac {3914701972 \sqrt {1-2 x} \sqrt {3+5 x}}{3754455705 (2+3 x)^{5/2}}+\frac {181941877952 \sqrt {1-2 x} \sqrt {3+5 x}}{26281189935 (2+3 x)^{3/2}}+\frac {12641611554328 \sqrt {1-2 x} \sqrt {3+5 x}}{183968329545 \sqrt {2+3 x}}-\frac {1085156 \sqrt {1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac {16636 \sqrt {1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac {256 \int \frac {\frac {112545140451525}{16}+\frac {355545324965475 x}{32}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{8278574829525} \\ & = -\frac {112817764 \sqrt {1-2 x} \sqrt {3+5 x}}{107270163 (2+3 x)^{7/2}}+\frac {3914701972 \sqrt {1-2 x} \sqrt {3+5 x}}{3754455705 (2+3 x)^{5/2}}+\frac {181941877952 \sqrt {1-2 x} \sqrt {3+5 x}}{26281189935 (2+3 x)^{3/2}}+\frac {12641611554328 \sqrt {1-2 x} \sqrt {3+5 x}}{183968329545 \sqrt {2+3 x}}-\frac {1085156 \sqrt {1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac {16636 \sqrt {1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac {190110479576 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{16724393595}+\frac {12641611554328 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{183968329545} \\ & = -\frac {112817764 \sqrt {1-2 x} \sqrt {3+5 x}}{107270163 (2+3 x)^{7/2}}+\frac {3914701972 \sqrt {1-2 x} \sqrt {3+5 x}}{3754455705 (2+3 x)^{5/2}}+\frac {181941877952 \sqrt {1-2 x} \sqrt {3+5 x}}{26281189935 (2+3 x)^{3/2}}+\frac {12641611554328 \sqrt {1-2 x} \sqrt {3+5 x}}{183968329545 \sqrt {2+3 x}}-\frac {1085156 \sqrt {1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac {16636 \sqrt {1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}-\frac {12641611554328 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16724393595 \sqrt {33}}-\frac {380220959152 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16724393595 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.94 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.38 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=\frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (853124799464729+8886579657279639 x+39676146370896231 x^2+98427465692862075 x^3+146528498784887100 x^4+130900492508039982 x^5+64974368463330312 x^6+13823602234657668 x^7\right )}{(2+3 x)^{15/2}}+4 i \sqrt {33} \left (1580201444291 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1627729064185 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{551904988635} \]
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Time = 1.32 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.20
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{23914845 \left (\frac {2}{3}+x \right )^{8}}+\frac {16058 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{103630995 \left (\frac {2}{3}+x \right )^{7}}-\frac {641434 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{379980315 \left (\frac {2}{3}+x \right )^{6}}+\frac {1813814 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{531972441 \left (\frac {2}{3}+x \right )^{5}}+\frac {1513936 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8688883203 \left (\frac {2}{3}+x \right )^{4}}+\frac {3914701972 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{101370304035 \left (\frac {2}{3}+x \right )^{3}}+\frac {181941877952 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{236530709415 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {25283223108656}{36793665909} x^{2}-\frac {12641611554328}{183968329545} x +\frac {12641611554328}{61322776515}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {16006419975328 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{3863334920445 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {25283223108656 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{3863334920445 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(374\) |
default | \(-\frac {2 \left (77419842517122564 x +785774579099136 \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 )-809063139476992 \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 )+139229433234128160 \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}-143355875026079520 \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}-2214305034568163712 x^{5}-4203787124900760138 x^{6}-3997525460519271384 x^{7}+500221362404680812 x^{3}-166810299141489255 x^{4}+304831834382285292 x^{2}-1990701860603882364 x^{8}-95570583350719680 \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}-414708067039730040 x^{9}+8250633080540928 \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}-8495162964508416 \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}+37127848862434176 \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}-38228233340287872 \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}+125306489910715344 \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}-129020287523471568 \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}+62653244955357672 \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}-64510143761735784 \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}+92819622156085440 \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}+13425695347576644 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{7} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-13823602234657668 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{7} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+7678123195182561\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{551904988635 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {15}{2}}}\) | \(789\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=\frac {2 \, {\left (135 \, {\left (13823602234657668 \, x^{7} + 64974368463330312 \, x^{6} + 130900492508039982 \, x^{5} + 146528498784887100 \, x^{4} + 98427465692862075 \, x^{3} + 39676146370896231 \, x^{2} + 8886579657279639 \, x + 853124799464729\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 107382958285054 \, \sqrt {-30} {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 284436259972380 \, \sqrt {-30} {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{24835724488575 \, {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {17}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {17}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{17/2}} \,d x \]
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