Integrand size = 28, antiderivative size = 191 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx=-\frac {429479 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{637875}+\frac {14318 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{70875}+\frac {362 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {4457606 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3189375}-\frac {429479 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3189375} \]
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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} \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx=-\frac {429479 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3189375}-\frac {4457606 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3189375}+\frac {2}{27} \sqrt {3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac {362 \sqrt {3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2835}+\frac {14318 \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}}{70875}-\frac {429479 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{637875} \]
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Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2}{27} \int \frac {\left (-51-\frac {181 x}{2}\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx \\ & = \frac {362 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {4 \int \frac {\left (-\frac {10167}{4}-\frac {21477 x}{4}\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx}{2835} \\ & = \frac {14318 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{70875}+\frac {362 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {8 \int \frac {\left (-\frac {91323}{4}-\frac {1288437 x}{8}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{212625} \\ & = -\frac {429479 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{637875}+\frac {14318 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{70875}+\frac {362 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {8 \int \frac {\frac {18881943}{16}+\frac {6686409 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1913625} \\ & = -\frac {429479 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{637875}+\frac {14318 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{70875}+\frac {362 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {4724269 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6378750}+\frac {4457606 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3189375} \\ & = -\frac {429479 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{637875}+\frac {14318 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{70875}+\frac {362 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{2835}+\frac {2}{27} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {4457606 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3189375}-\frac {429479 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3189375} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.45 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.54 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx=\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (343207+232110 x-1192500 x^2+945000 x^3\right )+4457606 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-4887085 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{9568125} \]
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Time = 1.34 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (4607823 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4457606 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-425250000 x^{6}+210600000 x^{5}+406188000 x^{4}-274683600 x^{3}-201359865 x^{2}+56926635 x +30888630\right )}{9568125 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(155\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {5158 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{14175}+\frac {343207 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{637875}+\frac {6293981 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{66976875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8915212 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{66976875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1060 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{567}+\frac {40 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(250\) |
risch | \(-\frac {\left (945000 x^{3}-1192500 x^{2}+232110 x +343207\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{637875 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {6293981 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{70166250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4457606 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{35083125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(257\) |
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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 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx=\frac {1}{637875} \, {\left (945000 \, x^{3} - 1192500 \, x^{2} + 232110 \, x + 343207\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {180704207}{861131250} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {4457606}{9568125} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx=\text {Timed out} \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2}} \,d x } \]
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\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2}} \,d x } \]
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Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{\sqrt {3\,x+2}} \,d x \]
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