Integrand size = 26, antiderivative size = 209 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=-\frac {395307 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}-\frac {4348377 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}} \]
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Time = 0.05 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 96, 95, 210} \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=-\frac {4348377 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}}+\frac {297 \sqrt {1-2 x} (5 x+3)^{7/2}}{160 (3 x+2)^4}+\frac {9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6}-\frac {1089 \sqrt {1-2 x} (5 x+3)^{5/2}}{2240 (3 x+2)^3}-\frac {11979 \sqrt {1-2 x} (5 x+3)^{3/2}}{12544 (3 x+2)^2}-\frac {395307 \sqrt {1-2 x} \sqrt {5 x+3}}{175616 (3 x+2)} \]
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Rule 95
Rule 96
Rule 98
Rule 210
Rubi steps \begin{align*} \text {integral}& = \frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9}{4} \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx \\ & = \frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297}{40} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx \\ & = \frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {3267}{320} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^4} \, dx \\ & = -\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {11979}{896} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx \\ & = -\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {395307 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{25088} \\ & = -\frac {395307 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {4348377 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{351232} \\ & = -\frac {395307 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {4348377 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{175616} \\ & = -\frac {395307 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}-\frac {4348377 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.43 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (64829376+482263920 x+1428134688 x^2+2108117296 x^3+1555340180 x^4+460633945 x^5\right )}{(2+3 x)^6}-21741885 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6146560} \]
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Time = 1.15 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.67
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (460633945 x^{5}+1555340180 x^{4}+2108117296 x^{3}+1428134688 x^{2}+482263920 x +64829376\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{878080 \left (2+3 x \right )^{6} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {4348377 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2458624 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(139\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (15849834165 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+63399336660 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+105665561100 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+6448875230 x^{5} \sqrt {-10 x^{2}-x +3}+93924943200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+21774762520 x^{4} \sqrt {-10 x^{2}-x +3}+46962471600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+29513642144 x^{3} \sqrt {-10 x^{2}-x +3}+12523325760 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +19993885632 x^{2} \sqrt {-10 x^{2}-x +3}+1391480640 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+6751694880 x \sqrt {-10 x^{2}-x +3}+907611264 \sqrt {-10 x^{2}-x +3}\right )}{12293120 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\) | \(346\) |
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Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=-\frac {21741885 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (460633945 \, x^{5} + 1555340180 \, x^{4} + 2108117296 \, x^{3} + 1428134688 \, x^{2} + 482263920 \, x + 64829376\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12293120 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
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\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{7}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.31 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\frac {272085}{307328} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{42 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {23 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{420 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {297 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1568 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {10989 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21952 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {489753 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{614656 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {6648345}{614656} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {4348377}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {5857731}{1229312} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {645909 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1229312 \, {\left (3 \, x + 2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (164) = 328\).
Time = 0.77 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.32 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\frac {4348377}{24586240} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {161051 \, \sqrt {10} {\left (27 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 42840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 27941760 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 6539187200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 940423680000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {46467993600000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {185871974400000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{87808 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^7} \,d x \]
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