Integrand size = 28, antiderivative size = 191 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=-\frac {1061}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2894 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835}-\frac {1061 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{2835} \]
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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 = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 159, 164, 114, 120} \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=-\frac {1061 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{2835}-\frac {2894 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835}-\frac {32}{63} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}+\frac {202}{63} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {1061}{567} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \]
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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} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {4}{315} \int \frac {\left (\frac {4495}{4}-\frac {7575 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (\frac {1125}{2}-\frac {79575 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{4725} \\ & = -\frac {1061}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {4 \int \frac {\frac {435525}{8}+\frac {108525 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{42525} \\ & = -\frac {1061}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2894 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2835}+\frac {11671 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5670} \\ & = -\frac {1061}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {202}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {32}{63} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2894 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835}-\frac {1061 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2835} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.80 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=\frac {2894 i \sqrt {33} (2+3 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-5 \left (3 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-200-1767 x-180 x^2+2700 x^3\right )+791 i \sqrt {33} (2+3 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{8505 (2+3 x)} \]
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Time = 1.26 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.79
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (3729 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-2894 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+405000 x^{5}+13500 x^{4}-389250 x^{3}-48405 x^{2}+76515 x +9000\right )}{8505 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(150\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {220 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{189}+\frac {149 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{567}+\frac {5807 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{59535 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5788 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{59535 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {100 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{63}-\frac {14 \left (-30 x^{2}-3 x +9\right )}{243 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.43 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=-\frac {1350 \, {\left (2700 \, x^{3} - 180 \, x^{2} - 1767 \, x - 200\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 194753 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 260460 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{765450 \, {\left (3 \, x + 2\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{3/2}} \,d x \]
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