Integrand size = 26, antiderivative size = 238 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=-\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}+\frac {84539611 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)^2}+\frac {8818415317 \sqrt {1-2 x} \sqrt {3+5 x}}{3252759552 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}-\frac {3735929329 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{120472576 \sqrt {7}} \]
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Time = 0.07 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {99, 154, 156, 12, 95, 210} \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=-\frac {3735929329 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{120472576 \sqrt {7}}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{1764 (3 x+2)^6}+\frac {8818415317 \sqrt {1-2 x} \sqrt {5 x+3}}{3252759552 (3 x+2)}+\frac {84539611 \sqrt {1-2 x} \sqrt {5 x+3}}{232339968 (3 x+2)^2}+\frac {2524471 \sqrt {1-2 x} \sqrt {5 x+3}}{41489280 (3 x+2)^3}+\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{20744640 (3 x+2)^4}-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{370440 (3 x+2)^5} \]
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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)^{5/2}}{21 (2+3 x)^7}+\frac {1}{21} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^7} \, dx \\ & = -\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {\int \frac {\left (-\frac {783}{4}-2760 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^6} \, dx}{2646} \\ & = -\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {\int \frac {-\frac {1154271}{8}-285690 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx}{277830} \\ & = -\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {\int \frac {\frac {8684811}{16}-\frac {16623405 x}{4}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{7779240} \\ & = -\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {\int \frac {\frac {4635547875}{32}-\frac {795208365 x}{4}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{163364040} \\ & = -\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}+\frac {84539611 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)^2}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {\int \frac {\frac {570867242085}{64}-\frac {133149887325 x}{16}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{2287096560} \\ & = -\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}+\frac {84539611 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)^2}+\frac {8818415317 \sqrt {1-2 x} \sqrt {3+5 x}}{3252759552 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {\int \frac {31774078943145}{128 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{16009675920} \\ & = -\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}+\frac {84539611 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)^2}+\frac {8818415317 \sqrt {1-2 x} \sqrt {3+5 x}}{3252759552 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {3735929329 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{240945152} \\ & = -\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}+\frac {84539611 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)^2}+\frac {8818415317 \sqrt {1-2 x} \sqrt {3+5 x}}{3252759552 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {3735929329 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{120472576} \\ & = -\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}+\frac {84539611 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)^2}+\frac {8818415317 \sqrt {1-2 x} \sqrt {3+5 x}}{3252759552 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}-\frac {3735929329 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{120472576 \sqrt {7}} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.40 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\frac {14641 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (335335888512+2987299350368 x+11077661454896 x^2+21898948566336 x^3+24351227238888 x^4+14445612678330 x^5+3571458203385 x^6\right )}{14641 (2+3 x)^7}-3827535 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{12649620480} \]
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Time = 1.12 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.61
method | result | size |
risch | \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (3571458203385 x^{6}+14445612678330 x^{5}+24351227238888 x^{4}+21898948566336 x^{3}+11077661454896 x^{2}+2987299350368 x +335335888512\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1807088640 \left (2+3 x \right )^{7} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {3735929329 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1686616064 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(144\) |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (122557161637845 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{7}+571933420976610 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}+1143866841953220 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+50000414847390 \sqrt {-10 x^{2}-x +3}\, x^{6}+1270963157725800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+202238577496620 x^{5} \sqrt {-10 x^{2}-x +3}+847308771817200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+340917181344432 x^{4} \sqrt {-10 x^{2}-x +3}+338923508726880 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+306585279928704 x^{3} \sqrt {-10 x^{2}-x +3}+75316335272640 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +155087260368544 x^{2} \sqrt {-10 x^{2}-x +3}+7172984311680 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+41822190905152 x \sqrt {-10 x^{2}-x +3}+4694702439168 \sqrt {-10 x^{2}-x +3}\right )}{25299240960 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{7}}\) | \(394\) |
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Time = 0.23 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=-\frac {56038939935 \, \sqrt {7} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (3571458203385 \, x^{6} + 14445612678330 \, x^{5} + 24351227238888 \, x^{4} + 21898948566336 \, x^{3} + 11077661454896 \, x^{2} + 2987299350368 \, x + 335335888512\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{25299240960 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\frac {3735929329}{1686616064} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {154377245}{90354432} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{147 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} - \frac {191 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4116 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {919 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{96040 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {72203 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{768320 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {2612695 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{6453888 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {92626347 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{60236288 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1142391613 \, \sqrt {-10 \, x^{2} - x + 3}}{361417728 \, {\left (3 \, x + 2\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (187) = 374\).
Time = 0.81 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.28 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\frac {3735929329}{16866160640} \, \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 {14641 \, \sqrt {10} {\left (765507 \, {\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 )}^{13} + 1428946400 \, {\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} + 1132297127360 \, {\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} - 334448649830400 \, {\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} - 85378328229376000 \, {\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} - 8754907317452800000 \, {\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 {368890400944128000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1475561603776512000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{180708864 \, {\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 )}^{7}} \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^8} \,d x \]
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