Integrand size = 28, antiderivative size = 158 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-\frac {556}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {184 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5 \sqrt {33}} \]
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Time = 0.04 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 157, 164, 114, 120} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=-\frac {184 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5 \sqrt {33}}-\frac {556}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {556 \sqrt {1-2 x} \sqrt {3 x+2}}{3 \sqrt {5 x+3}}-\frac {92 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{3 \sqrt {3 x+2} (5 x+3)^{3/2}} \]
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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} (3+5 x)^{3/2}}+\frac {2}{3} \int \frac {90-103 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx \\ & = \frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}-\frac {4}{99} \int \frac {\frac {7359}{2}-2277 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} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+\frac {8 \int \frac {47916+\frac {151371 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1089} \\ & = \frac {14 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+\frac {92}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {556}{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} (3+5 x)^{3/2}}-\frac {92 \sqrt {1-2 x} \sqrt {2+3 x}}{3 (3+5 x)^{3/2}}+\frac {556 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-\frac {556}{5} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {184 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.59 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {2 \sqrt {1-2 x} \left (1583+5144 x+4170 x^2\right )}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {4 i \left (1529 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1575 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{5 \sqrt {33}} \]
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Time = 1.33 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.39
method | result | size |
default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (1350 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-1390 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+810 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-834 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-41700 x^{3}-30590 x^{2}+9890 x +7915\right )}{15 \left (3+5 x \right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )}\) | \(219\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {-420 x^{2}-42 x +126}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {704 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{105 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1112 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{105 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{75 \left (x +\frac {3}{5}\right )^{2}}+\frac {-692 x^{2}-\frac {346}{3} x +\frac {692}{3}}{\sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(247\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (225 \, {\left (4170 \, x^{2} + 5144 \, x + 1583\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 4723 \, \sqrt {-30} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 12510 \, \sqrt {-30} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{675 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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