Integrand size = 28, antiderivative size = 222 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\frac {1353340 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35721}-\frac {62596 \sqrt {1-2 x} (3+5 x)^{3/2}}{3969 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {1844 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}-\frac {904798 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{35721}+\frac {270668 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{35721} \]
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Time = 0.07 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 159, 164, 114, 120} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\frac {270668 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{35721}-\frac {904798 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{35721}-\frac {1844 \sqrt {1-2 x} (5 x+3)^{5/2}}{567 (3 x+2)^{3/2}}+\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{63 (3 x+2)^{5/2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{21 (3 x+2)^{7/2}}-\frac {62596 \sqrt {1-2 x} (5 x+3)^{3/2}}{3969 \sqrt {3 x+2}}+\frac {1353340 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{35721} \]
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {4}{315} \int \frac {\left (-\frac {1415}{2}-\frac {3275 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {1844 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}+\frac {8 \int \frac {\left (\frac {19045}{4}-22200 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx}{2835} \\ & = -\frac {62596 \sqrt {1-2 x} (3+5 x)^{3/2}}{3969 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {1844 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}+\frac {16 \int \frac {\left (\frac {1656225}{8}-\frac {5075025 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{59535} \\ & = \frac {1353340 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35721}-\frac {62596 \sqrt {1-2 x} (3+5 x)^{3/2}}{3969 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {1844 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}-\frac {16 \int \frac {-\frac {2298225}{2}-\frac {33929925 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{535815} \\ & = \frac {1353340 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35721}-\frac {62596 \sqrt {1-2 x} (3+5 x)^{3/2}}{3969 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {1844 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}+\frac {904798 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{35721}-\frac {1488674 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{35721} \\ & = \frac {1353340 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35721}-\frac {62596 \sqrt {1-2 x} (3+5 x)^{3/2}}{3969 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{21 (2+3 x)^{7/2}}+\frac {74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{63 (2+3 x)^{5/2}}-\frac {1844 \sqrt {1-2 x} (3+5 x)^{5/2}}{567 (2+3 x)^{3/2}}-\frac {904798 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{35721}+\frac {270668 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{35721} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.74 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (2337569+11107911 x+17788023 x^2+9846603 x^3+396900 x^4\right )}{(2+3 x)^{7/2}}+i \sqrt {33} \left (452399 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-317065 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{107163} \]
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Time = 1.38 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.34
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19683 \left (\frac {2}{3}+x \right )^{4}}+\frac {74 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2187 \left (\frac {2}{3}+x \right )^{3}}-\frac {26882 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{45927 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {6509780}{35721} x^{2}-\frac {650978}{35721} x +\frac {650978}{11907}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {490288 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{750141 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1809596 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{750141 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {200 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{729}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(297\) |
default | \(-\frac {2 \left (8071569 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-12214773 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+16143138 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-24429546 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+10762092 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-16286364 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+2391576 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-3619192 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-11907000 x^{6}-296588790 x^{5}-559608399 x^{4}-297981972 x^{3}+56641404 x^{2}+92958492 x +21038121\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{107163 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\) | \(414\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.60 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\frac {270 \, {\left (396900 \, x^{4} + 9846603 \, x^{3} + 17788023 \, x^{2} + 11107911 \, x + 2337569\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 626303 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 40715910 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{4822335 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{9/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{9/2}} \,d x \]
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