Integrand size = 28, antiderivative size = 191 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{5/2}}+\frac {8516 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {595324 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}-\frac {595324 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}-\frac {18016 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{46305} \]
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Time = 0.05 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157, 164, 114, 120} \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=-\frac {18016 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{46305}-\frac {595324 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{21 (3 x+2)^{7/2}}+\frac {595324 \sqrt {5 x+3} \sqrt {1-2 x}}{46305 \sqrt {3 x+2}}+\frac {8516 \sqrt {5 x+3} \sqrt {1-2 x}}{6615 (3 x+2)^{3/2}}+\frac {82 \sqrt {5 x+3} \sqrt {1-2 x}}{315 (3 x+2)^{5/2}} \]
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {\left (-\frac {13}{2}-20 x\right ) \sqrt {1-2 x}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{5/2}}-\frac {4}{315} \int \frac {-\frac {433}{2}+\frac {415 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{5/2}}+\frac {8516 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}-\frac {8 \int \frac {-\frac {35417}{4}+\frac {10645 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{6615} \\ & = -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{5/2}}+\frac {8516 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {595324 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}-\frac {16 \int \frac {-\frac {471265}{4}-\frac {744155 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{46305} \\ & = -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{5/2}}+\frac {8516 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {595324 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}+\frac {99088 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{46305}+\frac {595324 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{46305} \\ & = -\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {82 \sqrt {1-2 x} \sqrt {3+5 x}}{315 (2+3 x)^{5/2}}+\frac {8516 \sqrt {1-2 x} \sqrt {3+5 x}}{6615 (2+3 x)^{3/2}}+\frac {595324 \sqrt {1-2 x} \sqrt {3+5 x}}{46305 \sqrt {2+3 x}}-\frac {595324 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305}-\frac {18016 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{46305} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.84 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.53 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=\frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (2510369+11095995 x+16342002 x^2+8036874 x^3\right )}{2 (2+3 x)^{7/2}}+i \sqrt {33} \left (148831 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-153335 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{138915} \]
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Time = 1.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.40
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{729 \left (\frac {2}{3}+x \right )^{4}}+\frac {34 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2835 \left (\frac {2}{3}+x \right )^{3}}+\frac {8516 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{59535 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {1190648}{9261} x^{2}-\frac {595324}{46305} x +\frac {595324}{15435}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {754024 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{972405 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1190648 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{972405 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(267\) |
default | \(-\frac {2 \left (7806942 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-8036874 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+15613884 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-16073748 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+10409256 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-10715832 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+2313168 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-2381296 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-241106220 x^{5}-514370682 x^{4}-309573990 x^{3}+38478963 x^{2}+92332848 x +22593321\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{138915 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\) | \(409\) |
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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 = 0.67 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=\frac {2 \, {\left (135 \, {\left (8036874 \, x^{3} + 16342002 \, x^{2} + 11095995 \, x + 2510369\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 5059657 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 13394790 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{6251175 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{9/2}} \,d x \]
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