Integrand size = 28, antiderivative size = 189 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {12101}{20} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {91}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 189, 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, 159, 164, 114, 120} \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {91}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {12101}{20} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {(3 x+2)^{3/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {137 \sqrt {3 x+2} (5 x+3)^{5/2}}{33 \sqrt {1-2 x}}-\frac {817}{66} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-91 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \]
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {77}{2}+60 x\right )}{(1-2 x)^{3/2}} \, dx \\ & = -\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {8033}{2}-\frac {12255 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1}{495} \int \frac {\sqrt {3+5 x} \left (\frac {1053855}{4}+405405 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\int \frac {-\frac {34129755}{4}-\frac {53909955 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4455} \\ & = -91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1001}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {12101}{20} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -91 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {817}{66} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {137 \sqrt {2+3 x} (3+5 x)^{5/2}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {12101}{20} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {91}{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 = 7.60 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.64 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {10 \sqrt {2+3 x} \sqrt {3+5 x} \left (957-2579 x+438 x^2+90 x^3\right )+12101 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-12465 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{60 (1-2 x)^{3/2}} \]
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Time = 1.38 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.26
method | result | size |
default | \(-\frac {\left (164538 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-169414 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-82269 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+84707 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+94500 x^{5}+579600 x^{4}-2087610 x^{3}-2241260 x^{2}+189630 x +401940\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}}{420 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(238\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-\frac {15 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4}-22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}+\frac {7661 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{210 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {12101 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{210 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {847 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{96 \left (x -\frac {1}{2}\right )^{2}}+\frac {-\frac {20735}{8} x^{2}-\frac {78793}{24} x -\frac {4147}{4}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(270\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {900 \, {\left (90 \, x^{3} + 438 \, x^{2} - 2579 \, x + 957\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 411167 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 1089090 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{5400 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
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Timed out. \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]
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