Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}+\frac {10312712 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {310208 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1715} \]
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Time = 0.07 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 157, 164, 114, 120} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\frac {310208 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1715}+\frac {10312712 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}-\frac {10312712 \sqrt {1-2 x} \sqrt {3 x+2}}{1029 \sqrt {5 x+3}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {12276 \sqrt {1-2 x}}{245 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {176 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {2 \sqrt {1-2 x}}{3 (3 x+2)^{7/2} \sqrt {5 x+3}} \]
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Rule 100
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {2}{21} \int \frac {154-231 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {4}{735} \int \frac {\frac {33649}{2}-23100 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx \\ & = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {8 \int \frac {1272579-\frac {2900205 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{15435} \\ & = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {217164255}{4}-33589710 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{108045} \\ & = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}-\frac {32 \int \frac {\frac {2827810755}{4}+\frac {4466693385 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1188495} \\ & = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}-\frac {1706144 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1715}-\frac {10312712 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1715} \\ & = \frac {2 \sqrt {1-2 x}}{3 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {176 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {12276 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {1706144 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {10312712 \sqrt {1-2 x} \sqrt {2+3 x}}{1029 \sqrt {3+5 x}}+\frac {10312712 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}+\frac {310208 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.33 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\frac {2 \left (-\frac {3 \sqrt {1-2 x} \left (130497191+793777840 x+1809835578 x^2+1833255216 x^3+696108060 x^4\right )}{(2+3 x)^{7/2} \sqrt {3+5 x}}-4 i \sqrt {33} \left (1289089 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1327865 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{5145} \]
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Time = 1.30 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.33
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81 \left (\frac {2}{3}+x \right )^{4}}-\frac {878 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{945 \left (\frac {2}{3}+x \right )^{3}}-\frac {67558 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2205 \left (\frac {2}{3}+x \right )^{2}}-\frac {7482962 \left (-30 x^{2}-3 x +9\right )}{5145 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {13057712 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{36015 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {20625424 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{36015 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {550 \left (-30 x^{2}-5 x +10\right )}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(295\) |
default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (135214596 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-139221612 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+270429192 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-278443224 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+180286128 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-185628816 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+40063584 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-41250848 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4176648360 x^{5}-8911207116 x^{4}-5359247820 x^{3}+666839694 x^{2}+1598350374 x +391491573\right )}{5145 \left (2+3 x \right )^{\frac {7}{2}} \left (10 x^{2}+x -3\right )}\) | \(409\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (135 \, {\left (696108060 \, x^{4} + 1833255216 \, x^{3} + 1809835578 \, x^{2} + 793777840 \, x + 130497191\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 87601166 \, \sqrt {-30} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 232036020 \, \sqrt {-30} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{231525 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{9/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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