Integrand size = 28, antiderivative size = 249 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}+\frac {4839325048 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}+\frac {145418632 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{10168235 \sqrt {33}} \]
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Time = 0.08 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {106, 157, 164, 114, 120} \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {145418632 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{10168235 \sqrt {33}}+\frac {4839325048 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {3 x+2}}{67110351 \sqrt {5 x+3}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {499564 \sqrt {1-2 x}}{1452605 (3 x+2)^{3/2} \sqrt {5 x+3}}-\frac {2206 \sqrt {1-2 x}}{207515 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {1616}{17787 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} \sqrt {5 x+3}} \]
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Rule 106
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2}{231} \int \frac {-\frac {269}{2}-135 x}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {4 \int \frac {\frac {55457}{4}+21210 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx}{17787} \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {8 \int \frac {\frac {429823}{4}+\frac {82725 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{622545} \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {32051607}{8}-\frac {16860285 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{13073445} \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {32 \int \frac {\frac {661979505}{4}-\frac {817979805 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{91514115} \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}-\frac {64 \int \frac {\frac {34464999645}{16}+\frac {27221203395 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1006655265} \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}-\frac {72709316 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{10168235}-\frac {4839325048 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{111850585} \\ & = \frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {1616}{17787 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2206 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {499564 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {72709316 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4839325048 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 \sqrt {3+5 x}}+\frac {4839325048 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}}+\frac {145418632 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{10168235 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.64 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {2 \left (-\frac {91855922241+53503915182 x-673871013766 x^2-559512908172 x^3+1263428429256 x^4+1306617762960 x^5}{(1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}-4 i \sqrt {33} \left (604915631 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-623092960 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{335551755} \]
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Time = 4.72 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.30
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {6 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{3}}-\frac {6 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35 \left (\frac {2}{3}+x \right )^{2}}-\frac {817326 \left (-30 x^{2}-3 x +9\right )}{84035 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {6127111048 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2348862285 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {9678650096 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{2348862285 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {16 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{871563 \left (x -\frac {1}{2}\right )^{2}}-\frac {17216 \left (-30 x^{2}-38 x -12\right )}{67110351 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {6250 \left (-30 x^{2}-5 x +10\right )}{1331 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(323\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \left (43553925432 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-42299110656 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+36294937860 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-35249258880 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9678650096 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+9399802368 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9678650096 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+9399802368 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+1306617762960 x^{5}+1263428429256 x^{4}-559512908172 x^{3}-673871013766 x^{2}+53503915182 x +91855922241\right )}{335551755 \left (2+3 x \right )^{\frac {5}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\) | \(406\) |
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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 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (45 \, {\left (1306617762960 \, x^{5} + 1263428429256 \, x^{4} - 559512908172 \, x^{3} - 673871013766 \, x^{2} + 53503915182 \, x + 91855922241\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 41103880264 \, \sqrt {-30} {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 108884813580 \, \sqrt {-30} {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{15099828975 \, {\left (540 \, x^{6} + 864 \, x^{5} + 99 \, x^{4} - 425 \, x^{3} - 154 \, x^{2} + 52 \, x + 24\right )}} \]
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Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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