Integrand size = 28, antiderivative size = 129 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {136 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+\frac {136}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {4}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 157, 164, 114, 120} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {4}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {136}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {136 \sqrt {1-2 x} \sqrt {3 x+2}}{3 \sqrt {5 x+3}}+\frac {14 \sqrt {1-2 x}}{3 \sqrt {3 x+2} \sqrt {5 x+3}} \]
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Rule 100
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {2}{3} \int \frac {55-33 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx \\ & = \frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {136 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-\frac {4}{33} \int \frac {\frac {1419}{2}+1122 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = \frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {136 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-\frac {22}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {136}{5} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {136 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+\frac {136}{5} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {4}{5} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 5.90 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {1-2 x} (43+68 x)}{\sqrt {2+3 x} \sqrt {3+5 x}}-\frac {4}{5} i \sqrt {\frac {11}{3}} \left (34 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-35 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \]
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Time = 1.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (66 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-68 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-2040 x^{2}-270 x +645\right )}{15 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(135\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (15-30 x \right ) \left (\frac {43}{15}+\frac {68 x}{15}\right )}{\sqrt {\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right ) \left (15-30 x \right )}}-\frac {172 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{105 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {272 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{105 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(195\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (675 \, {\left (68 \, x + 43\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1153 \, \sqrt {-30} {\left (15 \, x^{2} + 19 \, x + 6\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 3060 \, \sqrt {-30} {\left (15 \, x^{2} + 19 \, x + 6\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{675 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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