Integrand size = 28, antiderivative size = 191 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {62092 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2835}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {8256877 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{56700}-\frac {62092 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{14175} \]
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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 = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {103, 159, 164, 114, 120} \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {62092 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{14175}-\frac {8256877 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{56700}-\frac {1}{9} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {62092 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{2835} \]
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Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {1}{9} \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {173}{2}+135 x\right )}{\sqrt {1-2 x}} \, dx \\ & = -\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{315} \int \frac {\left (-\frac {18455}{2}-\frac {28155 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = -\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {\int \frac {\sqrt {3+5 x} \left (\frac {2421135}{4}+931380 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{4725} \\ & = -\frac {62092 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2835}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {\int \frac {-\frac {78409965}{4}-\frac {123853155 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{42525} \\ & = -\frac {62092 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2835}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {341506 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{14175}+\frac {8256877 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{56700} \\ & = -\frac {62092 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2835}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {8256877 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{56700}-\frac {62092 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{14175} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.56 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.54 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\frac {-30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (208073+212175 x+148950 x^2+47250 x^3\right )+8256877 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-8505245 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{170100} \]
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Time = 1.35 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (8019231 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-8256877 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+42525000 x^{6}+166657500 x^{5}+283810500 x^{4}+293881950 x^{3}+72202620 x^{2}-81886830 x -37453140\right )}{170100 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(155\) |
risch | \(\frac {\left (47250 x^{3}+148950 x^{2}+212175 x +208073\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{5670 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {5227331 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{623700 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8256877 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{623700 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-\frac {4715 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{126}-\frac {208073 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5670}+\frac {5227331 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{595350 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8256877 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{595350 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1655 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{63}-\frac {25 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(262\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.34 \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {1}{5670} \, {\left (47250 \, x^{3} + 148950 \, x^{2} + 212175 \, x + 208073\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {280551619}{15309000} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {8256877}{170100} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
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Timed out. \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\text {Timed out} \]
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\[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{\sqrt {-2 \, x + 1}} \,d x } \]
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\[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{\sqrt {-2 \, x + 1}} \,d x } \]
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Timed out. \[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \]
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