Integrand size = 26, antiderivative size = 178 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=-\frac {77269 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)}-\frac {871 \sqrt {1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {100}{243} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {1922677 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{762048 \sqrt {7}} \]
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Time = 0.05 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {99, 154, 163, 56, 222, 95, 210} \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {100}{243} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {1922677 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{762048 \sqrt {7}}+\frac {181 \sqrt {1-2 x} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac {871 \sqrt {1-2 x} (5 x+3)^{3/2}}{6048 (3 x+2)^2}-\frac {77269 \sqrt {1-2 x} \sqrt {5 x+3}}{254016 (3 x+2)} \]
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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)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {1}{12} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx \\ & = -\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}-\frac {1}{108} \int \frac {\left (-\frac {1511}{4}-240 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx \\ & = -\frac {871 \sqrt {1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}-\frac {\int \frac {\left (-\frac {166869}{8}-16800 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{4536} \\ & = -\frac {77269 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)}-\frac {871 \sqrt {1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}-\frac {\int \frac {-\frac {8194677}{16}-588000 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{95256} \\ & = -\frac {77269 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)}-\frac {871 \sqrt {1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1922677 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1524096}+\frac {500}{243} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {77269 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)}-\frac {871 \sqrt {1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {1922677 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{762048}+\frac {1}{243} \left (200 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right ) \\ & = -\frac {77269 \sqrt {1-2 x} \sqrt {3+5 x}}{254016 (2+3 x)}-\frac {871 \sqrt {1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac {181 \sqrt {1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac {100}{243} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {1922677 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{762048 \sqrt {7}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} \left (2583760+13434180 x+23185560 x^2+13290147 x^3\right )}{(2+3 x)^4}-2195200 \sqrt {10} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-1922677 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5334336} \]
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Time = 1.15 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (13290147 x^{3}+23185560 x^{2}+13434180 x +2583760\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{254016 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {50 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{243}+\frac {1922677 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{10668672}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(142\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (155736837 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+177811200 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{4}+415298232 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+474163200 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}+415298232 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+474163200 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}+558186174 x^{3} \sqrt {-10 x^{2}-x +3}+184576992 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +210739200 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +973793520 x^{2} \sqrt {-10 x^{2}-x +3}+30762832 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+35123200 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+564235560 x \sqrt {-10 x^{2}-x +3}+108517920 \sqrt {-10 x^{2}-x +3}\right )}{10668672 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\) | \(315\) |
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Time = 0.24 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=-\frac {1922677 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 2195200 \, \sqrt {10} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \, {\left (13290147 \, x^{3} + 23185560 \, x^{2} + 13434180 \, x + 2583760\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{10668672 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
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\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{5}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.11 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {27065}{148176} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{28 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {169 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1176 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {5413 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{32928 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {528205}{296352} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {50}{243} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {1922677}{10668672} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {802877}{1778112} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {3667 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{197568 \, {\left (3 \, x + 2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (136) = 272\).
Time = 0.65 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.44 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\frac {1922677}{106686720} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {50}{243} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (77269 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 81002040 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 31057924800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {8580356288000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {34321425152000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{127008 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^5} \,d x \]
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