Integrand size = 26, antiderivative size = 207 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=-\frac {3248687 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)}-\frac {32453 \sqrt {1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}-\frac {200}{729} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {109715471 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4572288 \sqrt {7}} \]
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Time = 0.06 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {99, 154, 163, 56, 222, 95, 210} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=-\frac {200}{729} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {109715471 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{4572288 \sqrt {7}}+\frac {2543 \sqrt {1-2 x} (5 x+3)^{5/2}}{1296 (3 x+2)^3}+\frac {37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{72 (3 x+2)^4}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}-\frac {32453 \sqrt {1-2 x} (5 x+3)^{3/2}}{36288 (3 x+2)^2}-\frac {3248687 \sqrt {1-2 x} \sqrt {5 x+3}}{1524096 (3 x+2)} \]
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Rule 56
Rule 95
Rule 99
Rule 154
Rule 163
Rule 210
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx \\ & = -\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}-\frac {1}{180} \int \frac {\left (-\frac {5305}{4}-400 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx \\ & = -\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}+\frac {\int \frac {\left (\frac {149465}{8}-2400 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx}{1620} \\ & = -\frac {32453 \sqrt {1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}+\frac {\int \frac {\left (\frac {14451435}{16}-168000 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{68040} \\ & = -\frac {3248687 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)}-\frac {32453 \sqrt {1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}+\frac {\int \frac {\frac {423137355}{32}-5880000 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1428840} \\ & = -\frac {3248687 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)}-\frac {32453 \sqrt {1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}-\frac {1000}{729} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {109715471 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{9144576} \\ & = -\frac {3248687 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)}-\frac {32453 \sqrt {1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}+\frac {109715471 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{4572288}-\frac {1}{729} \left (400 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right ) \\ & = -\frac {3248687 \sqrt {1-2 x} \sqrt {3+5 x}}{1524096 (2+3 x)}-\frac {32453 \sqrt {1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac {37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac {2543 \sqrt {1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}-\frac {200}{729} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {109715471 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4572288 \sqrt {7}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (180761312+1044006792 x+2146957188 x^2+1809469170 x^3+490413015 x^4\right )}{(2+3 x)^5}+43904000 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-548577355 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{160030080} \]
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Time = 1.51 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.71
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (490413015 x^{4}+1809469170 x^{3}+2146957188 x^{2}+1044006792 x +180761312\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{7620480 \left (2+3 x \right )^{5} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (\frac {100 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{729}-\frac {109715471 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{64012032}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(148\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (133304297265 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}-10668672000 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{5}+444347657550 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-35562240000 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}+592463543400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-47416320000 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}+20597346630 x^{4} \sqrt {-10 x^{2}-x +3}+394975695600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-31610880000 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+75997705140 x^{3} \sqrt {-10 x^{2}-x +3}+131658565200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -10536960000 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +90172201896 x^{2} \sqrt {-10 x^{2}-x +3}+17554475360 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1404928000 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+43848285264 x \sqrt {-10 x^{2}-x +3}+7591975104 \sqrt {-10 x^{2}-x +3}\right )}{320060160 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\) | \(377\) |
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Time = 0.24 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=-\frac {548577355 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 43904000 \, \sqrt {10} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \, {\left (490413015 \, x^{4} + 1809469170 \, x^{3} + 2146957188 \, x^{2} + 1044006792 \, x + 180761312\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{320060160 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\text {Timed out} \]
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Time = 0.31 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.29 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\frac {44881}{691488} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{35 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {333 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{1960 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {6347 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{27440 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {44881 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{768320 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {3156205}{1382976} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {52017151}{24893568} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {9235489 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{13829760 \, {\left (3 \, x + 2\right )}} + \frac {17832215}{1778112} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {100}{729} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {109715471}{64012032} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {49508071}{10668672} \, \sqrt {-10 \, x^{2} - x + 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (159) = 318\).
Time = 0.74 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.38 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\frac {109715471}{640120320} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {100}{729} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (3248687 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 4238260880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 2165236899840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 364930179712000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {12258004702720000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {49032018810880000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{762048 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^6} \,d x \]
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