Integrand size = 28, antiderivative size = 191 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\frac {5264}{243} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{3/2}}-\frac {316 \sqrt {1-2 x} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}-\frac {19174 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1215}+\frac {5264 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1215} \]
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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, 159, 164, 114, 120} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\frac {5264 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1215}-\frac {19174 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1215}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{15 (3 x+2)^{5/2}}+\frac {46 (5 x+3)^{3/2} (1-2 x)^{3/2}}{27 (3 x+2)^{3/2}}-\frac {316 (5 x+3)^{3/2} \sqrt {1-2 x}}{27 \sqrt {3 x+2}}+\frac {5264}{243} \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x} \]
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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)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {2}{15} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{3/2}}-\frac {4}{135} \int \frac {\left (-\frac {915}{2}-\frac {1965 x}{2}\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx \\ & = -\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{3/2}}-\frac {316 \sqrt {1-2 x} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}+\frac {8}{405} \int \frac {\left (\frac {6705}{4}-9870 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {5264}{243} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{3/2}}-\frac {316 \sqrt {1-2 x} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}-\frac {8 \int \frac {-\frac {42855}{4}-\frac {143805 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3645} \\ & = \frac {5264}{243} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{3/2}}-\frac {316 \sqrt {1-2 x} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}+\frac {19174 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1215}-\frac {28952 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1215} \\ & = \frac {5264}{243} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {46 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{3/2}}-\frac {316 \sqrt {1-2 x} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}-\frac {19174 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1215}+\frac {5264 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1215} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.55 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (25927+83412 x+68913 x^2+2700 x^3\right )}{(2+3 x)^{5/2}}+i \sqrt {33} \left (9587 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-6955 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{3645} \]
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Time = 1.30 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.43
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {3248 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{10935 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {28228}{243} x^{2}-\frac {14114}{1215} x +\frac {14114}{405}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {11428 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{25515 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {38348 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{25515 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {40 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{243}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{10935 \left (\frac {2}{3}+x \right )^{3}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(273\) |
default | \(-\frac {2 \left (413127 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-603981 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+550836 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-805308 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+183612 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-268436 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-567000 x^{5}-14528430 x^{4}-18793593 x^{3}-2854803 x^{2}+4710489 x +1633401\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{25515 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {5}{2}}}\) | \(319\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.59 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\frac {270 \, {\left (2700 \, x^{3} + 68913 \, x^{2} + 83412 \, x + 25927\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 36629 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 862830 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{164025 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{7/2}} \,d x \]
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