Integrand size = 28, antiderivative size = 160 \[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=-\frac {2252 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4725}-\frac {31}{525} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {148831 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47250}-\frac {2252 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{23625} \]
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Time = 0.04 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {103, 159, 164, 114, 120} \[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=-\frac {2252 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{23625}-\frac {148831 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47250}+\frac {2}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {31}{525} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {2252 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{4725} \]
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Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {2}{35} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2}{35} \int \frac {\left (-\frac {23}{2}-\frac {31 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -\frac {31}{525} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{525} \int \frac {\sqrt {3+5 x} \left (\frac {2907}{4}+1126 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -\frac {2252 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4725}-\frac {31}{525} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2 \int \frac {-\frac {94253}{4}-\frac {148831 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4725} \\ & = -\frac {2252 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4725}-\frac {31}{525} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {12386 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{23625}+\frac {148831 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{47250} \\ & = -\frac {2252 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4725}-\frac {31}{525} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {148831 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47250}-\frac {2252 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{23625} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.96 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.61 \[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-659+6705 x+6750 x^2\right )+148831 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-153335 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{141750} \]
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Time = 1.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (144573 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-148831 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-6075000 x^{5}-10692000 x^{4}-2615850 x^{3}+3077760 x^{2}+1068510 x -118620\right )}{141750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(150\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {149 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{105}-\frac {659 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4725}+\frac {94253 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{496125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {148831 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{496125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {10 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(228\) |
risch | \(-\frac {\left (6750 x^{2}+6705 x -659\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{4725 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {94253 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{519750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {148831 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{519750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(252\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.37 \[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\frac {1}{4725} \, {\left (6750 \, x^{2} + 6705 \, x - 659\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {5059657}{12757500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {148831}{141750} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
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\[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\int \sqrt {1 - 2 x} \sqrt {3 x + 2} \left (5 x + 3\right )^{\frac {3}{2}}\, dx \]
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\[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \,d x } \]
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\[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \,d x } \]
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Timed out. \[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\int \sqrt {1-2\,x}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{3/2} \,d x \]
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