Integrand size = 28, antiderivative size = 191 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {106 \sqrt {1-2 x} (2+3 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {2264 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}+\frac {1558}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {1973 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}-\frac {8366 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{15625} \]
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Time = 0.06 (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} \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=-\frac {8366 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{15625}+\frac {1973 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}-\frac {106 \sqrt {1-2 x} (3 x+2)^{5/2}}{25 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}+\frac {1558}{625} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {2264 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{3125} \]
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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)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (\frac {3}{2}-24 x\right ) \sqrt {1-2 x} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {106 \sqrt {1-2 x} (2+3 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {4}{75} \int \frac {\left (306-\frac {2337 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {106 \sqrt {1-2 x} (2+3 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {1558}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {4 \int \frac {\sqrt {2+3 x} \left (-\frac {2775}{4}+5094 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1875} \\ & = -\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {106 \sqrt {1-2 x} (2+3 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {2264 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}+\frac {1558}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {4 \int \frac {\frac {5967}{2}-\frac {17757 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{28125} \\ & = -\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {106 \sqrt {1-2 x} (2+3 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {2264 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}+\frac {1558}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {1973 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{15625}+\frac {12549 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{15625} \\ & = -\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {106 \sqrt {1-2 x} (2+3 x)^{5/2}}{25 \sqrt {3+5 x}}+\frac {2264 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3125}+\frac {1558}{625} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {1973 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625}-\frac {8366 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{15625} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.44 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.54 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=\frac {-\frac {110 \sqrt {1-2 x} \sqrt {2+3 x} \left (106-2975 x-1050 x^2+6750 x^3\right )}{(3+5 x)^{3/2}}-21703 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-3395 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{515625} \]
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Time = 1.30 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.20
method | result | size |
default | \(-\frac {\left (1455 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+9865 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+873 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+5919 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+405000 x^{5}+4500 x^{4}-324000 x^{3}-2390 x^{2}+60560 x -2120\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}}{46875 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(229\) |
elliptic | \(-\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {2+3 x}\, \sqrt {1-2 x}\, \left (-\frac {36 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{625}+\frac {244 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3125}+\frac {884 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{109375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {3946 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{328125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{46875 \left (x +\frac {3}{5}\right )^{2}}-\frac {446 \left (-30 x^{2}-5 x +10\right )}{9375 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\) | \(269\) |
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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 = 0.51 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=-\frac {900 \, {\left (6750 \, x^{3} - 1050 \, x^{2} - 2975 \, x + 106\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 164719 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 177570 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{4218750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \]
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