Integrand size = 28, antiderivative size = 187 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {45040 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 (3+5 x)^{3/2}}+\frac {2976620 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 \sqrt {3+5 x}}-\frac {595324 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}}-\frac {18016 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5929 \sqrt {33}} \]
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Time = 0.05 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {106, 157, 164, 114, 120} \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=-\frac {18016 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5929 \sqrt {33}}-\frac {595324 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}}+\frac {2976620 \sqrt {1-2 x} \sqrt {3 x+2}}{195657 \sqrt {5 x+3}}-\frac {45040 \sqrt {1-2 x} \sqrt {3 x+2}}{17787 (5 x+3)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {4}{77 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}} \]
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Rule 106
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {2}{77} \int \frac {-\frac {131}{2}-75 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx \\ & = \frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {4}{539} \int \frac {-\frac {1415}{2}+\frac {1395 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx \\ & = \frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {45040 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 (3+5 x)^{3/2}}+\frac {8 \int \frac {-\frac {108295}{4}+16890 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{17787} \\ & = \frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {45040 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 (3+5 x)^{3/2}}+\frac {2976620 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 \sqrt {3+5 x}}-\frac {16 \int \frac {-\frac {1413795}{4}-\frac {2232465 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{195657} \\ & = \frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {45040 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 (3+5 x)^{3/2}}+\frac {2976620 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 \sqrt {3+5 x}}+\frac {9008 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5929}+\frac {595324 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{65219} \\ & = \frac {4}{77 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {186 \sqrt {1-2 x}}{539 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {45040 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 (3+5 x)^{3/2}}+\frac {2976620 \sqrt {1-2 x} \sqrt {2+3 x}}{195657 \sqrt {3+5 x}}-\frac {595324 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}}-\frac {18016 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5929 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.39 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {8473261+10598372 x-32744810 x^2-44649300 x^3}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}+2 i \sqrt {33} \left (148831 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-153335 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{195657} \]
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Time = 1.36 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.17
method | result | size |
default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (1445730 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1488310 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+867438 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-892986 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-44649300 x^{3}-32744810 x^{2}+10598372 x +8473261\right )}{195657 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\) | \(219\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-18-30 x \right ) \left (\frac {107843}{391314}-\frac {35929 x}{65219}\right )}{\sqrt {\left (x^{2}+\frac {1}{6} x -\frac {1}{3}\right ) \left (-18-30 x \right )}}-\frac {10 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{363 \left (x +\frac {3}{5}\right )^{2}}+\frac {-\frac {77500}{1331} x^{2}-\frac {38750}{3993} x +\frac {77500}{3993}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {754024 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{1369599 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1190648 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{1369599 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(247\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (44649300 \, x^{3} + 32744810 \, x^{2} - 10598372 \, x - 8473261\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 5059657 \, \sqrt {-30} {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 13394790 \, \sqrt {-30} {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{8804565 \, {\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )}} \]
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Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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