Integrand size = 26, antiderivative size = 209 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=-\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}+\frac {1460201 \sqrt {1-2 x} \sqrt {3+5 x}}{2370816 (2+3 x)^2}+\frac {152571047 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}-\frac {64645339 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1229312 \sqrt {7}} \]
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Time = 0.07 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {99, 154, 156, 12, 95, 210} \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=-\frac {64645339 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1229312 \sqrt {7}}-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac {152571047 \sqrt {1-2 x} \sqrt {5 x+3}}{33191424 (3 x+2)}+\frac {1460201 \sqrt {1-2 x} \sqrt {5 x+3}}{2370816 (3 x+2)^2}+\frac {42461 \sqrt {1-2 x} \sqrt {5 x+3}}{423360 (3 x+2)^3}+\frac {4619 \sqrt {1-2 x} \sqrt {5 x+3}}{211680 (3 x+2)^4}-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{3780 (3 x+2)^5} \]
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Rule 12
Rule 95
Rule 99
Rule 154
Rule 156
Rule 210
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {1}{18} \int \frac {\left (\frac {9}{2}-20 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^6} \, dx \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {\int \frac {-\frac {2087}{4}-1360 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx}{1890} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {\int \frac {\frac {112467}{8}-\frac {69285 x}{2}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{52920} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {\int \frac {\frac {27328875}{16}-\frac {4458405 x}{2}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{1111320} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}+\frac {1460201 \sqrt {1-2 x} \sqrt {3+5 x}}{2370816 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {\int \frac {\frac {3295705245}{32}-\frac {766605525 x}{8}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{15558480} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}+\frac {1460201 \sqrt {1-2 x} \sqrt {3+5 x}}{2370816 (2+3 x)^2}+\frac {152571047 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {\int \frac {183269536065}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{108909360} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}+\frac {1460201 \sqrt {1-2 x} \sqrt {3+5 x}}{2370816 (2+3 x)^2}+\frac {152571047 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {64645339 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2458624} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}+\frac {1460201 \sqrt {1-2 x} \sqrt {3+5 x}}{2370816 (2+3 x)^2}+\frac {152571047 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {64645339 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1229312} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}+\frac {1460201 \sqrt {1-2 x} \sqrt {3+5 x}}{2370816 (2+3 x)^2}+\frac {152571047 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}-\frac {64645339 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1229312 \sqrt {7}} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\frac {1331 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (2906375616+21497808880 x+63585046048 x^2+94045700016 x^3+69576897780 x^4+20597091345 x^5\right )}{1331 (2+3 x)^6}-728535 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{129077760} \]
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Time = 3.62 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.67
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (20597091345 x^{5}+69576897780 x^{4}+94045700016 x^{3}+63585046048 x^{2}+21497808880 x +2906375616\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{18439680 \left (2+3 x \right )^{6} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {64645339 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{17210368 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(139\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (706896781965 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+2827587127860 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+4712645213100 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+288359278830 x^{5} \sqrt {-10 x^{2}-x +3}+4189017967200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+974076568920 x^{4} \sqrt {-10 x^{2}-x +3}+2094508983600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1316639800224 x^{3} \sqrt {-10 x^{2}-x +3}+558535728960 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +890190644672 x^{2} \sqrt {-10 x^{2}-x +3}+62059525440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+300969324320 x \sqrt {-10 x^{2}-x +3}+40689258624 \sqrt {-10 x^{2}-x +3}\right )}{258155520 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\) | \(346\) |
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Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=-\frac {969680085 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (20597091345 \, x^{5} + 69576897780 \, x^{4} + 94045700016 \, x^{3} + 63585046048 \, x^{2} + 21497808880 \, x + 2906375616\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{258155520 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
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\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{7}}\, dx \]
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Time = 0.33 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\frac {64645339}{17210368} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2671295}{921984} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{42 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {29 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{980 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {1273 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{7840 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {45245 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{65856 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {1602777 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{614656 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {19767583 \, \sqrt {-10 \, x^{2} - x + 3}}{3687936 \, {\left (3 \, x + 2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (164) = 328\).
Time = 0.67 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.32 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\frac {64645339}{172103680} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (145707 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 231188440 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 144245619840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 41365512115200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 5067855403520000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {250767109017600000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1003068436070400000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1843968 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^7} \,d x \]
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