Integrand size = 28, antiderivative size = 160 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{3/2}}{5 \sqrt {3+5 x}}+\frac {458}{625} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {24}{125} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {169 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125}-\frac {496 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3125} \]
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Time = 0.04 (sec) , antiderivative size = 160, 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 = {99, 159, 164, 114, 120} \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx=-\frac {496 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3125}-\frac {169 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125}-\frac {24}{125} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}-\frac {2 (1-2 x)^{3/2} (3 x+2)^{3/2}}{5 \sqrt {5 x+3}}+\frac {458}{625} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2} \]
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Rule 99
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{3/2} (2+3 x)^{3/2}}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {\left (-\frac {3}{2}-18 x\right ) \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2} (2+3 x)^{3/2}}{5 \sqrt {3+5 x}}-\frac {24}{125} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {4}{375} \int \frac {\left (\frac {675}{4}-\frac {2061 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2} (2+3 x)^{3/2}}{5 \sqrt {3+5 x}}+\frac {458}{625} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {24}{125} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {4 \int \frac {-\frac {5823}{4}-\frac {1521 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5625} \\ & = -\frac {2 (1-2 x)^{3/2} (2+3 x)^{3/2}}{5 \sqrt {3+5 x}}+\frac {458}{625} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {24}{125} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {169 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3125}+\frac {2728 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3125} \\ & = -\frac {2 (1-2 x)^{3/2} (2+3 x)^{3/2}}{5 \sqrt {3+5 x}}+\frac {458}{625} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {24}{125} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {169 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125}-\frac {496 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3125} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx=\frac {\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \left (77+130 x-150 x^2\right )}{\sqrt {3+5 x}}+169 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-665 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{9375} \]
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Time = 1.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.91
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (627 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-169 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+27000 x^{4}-18900 x^{3}-26760 x^{2}+5490 x +4620\right )}{9375 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(145\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {12 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{125}+\frac {88 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{625}+\frac {1294 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{65625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {338 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{65625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {22 \left (-30 x^{2}-5 x +10\right )}{625 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(234\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.49 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx=-\frac {2700 \, {\left (150 \, x^{2} - 130 \, x - 77\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 54343 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 15210 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{843750 \, {\left (5 \, x + 3\right )}} \]
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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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\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \]
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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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