Integrand size = 28, antiderivative size = 249 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=-\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}-\frac {1305025844 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {37904696 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{47647845 \sqrt {33}} \]
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Time = 0.07 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157, 164, 114, 120} \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=-\frac {37904696 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {1305025844 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{2079 (3 x+2)^{9/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {5 x+3}}{524126295 \sqrt {3 x+2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {5 x+3}}{74875185 (3 x+2)^{3/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {5 x+3}}{10696455 (3 x+2)^{5/2}}-\frac {13022 \sqrt {1-2 x} \sqrt {5 x+3}}{305613 (3 x+2)^{7/2}} \]
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{11/2}} \, dx \\ & = -\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {4 \int \frac {\left (-\frac {189}{4}-\frac {5025 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx}{6237} \\ & = -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {8 \int \frac {-\frac {676497}{8}-185700 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{916839} \\ & = -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {16 \int \frac {\frac {1286433}{2}-\frac {14125635 x}{8}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{32089365} \\ & = -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {32 \int \frac {\frac {687512943}{16}-\frac {109221165 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{673876665} \\ & = -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {64 \int \frac {\frac {2319498765}{4}+\frac {14681540745 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4717136655} \\ & = -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {18952348 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{47647845}+\frac {1305025844 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{524126295} \\ & = -\frac {13022 \sqrt {1-2 x} \sqrt {3+5 x}}{305613 (2+3 x)^{7/2}}+\frac {627806 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac {19417096 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac {1305025844 \sqrt {1-2 x} \sqrt {3+5 x}}{524126295 \sqrt {2+3 x}}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}-\frac {1305025844 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}}-\frac {37904696 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47647845 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.45 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (21813966691+162787885893 x+484598540169 x^2+719808574005 x^3+534040213536 x^4+158560640046 x^5\right )}{2 (2+3 x)^{11/2}}+i \sqrt {33} \left (326256461 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-335732635 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{1572378885} \]
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Time = 1.34 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.27
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{6}}+\frac {538 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1515591 \left (\frac {2}{3}+x \right )^{5}}-\frac {93382 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{24754653 \left (\frac {2}{3}+x \right )^{4}}+\frac {627806 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{288804285 \left (\frac {2}{3}+x \right )^{3}}+\frac {19417096 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{673876665 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {2610051688}{104825259} x^{2}-\frac {1305025844}{524126295} x +\frac {1305025844}{174708765}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {1649421344 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{11006652195 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2610051688 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{11006652195 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(315\) |
default | \(-\frac {2 \left (153842285718 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-158560640046 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+512807619060 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-528535466820 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+683743492080 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-704713955760 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+455828994720 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-469809303840 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+151942998240 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-156603101280 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-4756819201380 x^{7}+20259066432 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-20880413504 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-16496888326218 x^{6}-21769332100344 x^{5}-11891020005261 x^{4}+140844968748 x^{3}+3218604203112 x^{2}+1399649072964 x +196325700219\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{1572378885 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\) | \(599\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\frac {2 \, {\left (135 \, {\left (158560640046 \, x^{5} + 534040213536 \, x^{4} + 719808574005 \, x^{3} + 484598540169 \, x^{2} + 162787885893 \, x + 21813966691\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 11052091517 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 29363081490 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{70757049825 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{13/2}} \,d x \]
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