Integrand size = 28, antiderivative size = 125 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=\frac {2 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}+\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{1617 \sqrt {1-2 x}}+\frac {31 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}}+\frac {4 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{49 \sqrt {33}} \]
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Time = 0.03 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {101, 157, 164, 114, 120} \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=\frac {4 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{49 \sqrt {33}}+\frac {31 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}}+\frac {62 \sqrt {3 x+2} \sqrt {5 x+3}}{1617 \sqrt {1-2 x}}+\frac {2 \sqrt {3 x+2} \sqrt {5 x+3}}{21 (1-2 x)^{3/2}} \]
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Rule 101
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}-\frac {2}{21} \int \frac {-4-\frac {15 x}{2}}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = \frac {2 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}+\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{1617 \sqrt {1-2 x}}+\frac {4 \int \frac {-\frac {345}{4}-\frac {465 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1617} \\ & = \frac {2 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}+\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{1617 \sqrt {1-2 x}}-\frac {2}{49} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {31}{539} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx \\ & = \frac {2 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}+\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{1617 \sqrt {1-2 x}}+\frac {31 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}}+\frac {4 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{49 \sqrt {33}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.74 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=-\frac {4 \sqrt {2+3 x} \sqrt {3+5 x} (-54+31 x)-31 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+35 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{1617 (1-2 x)^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(218\) vs. \(2(93)=186\).
Time = 4.63 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.75
method | result | size |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{42 \left (x -\frac {1}{2}\right )^{2}}-\frac {31 \left (-30 x^{2}-38 x -12\right )}{1617 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {46 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{11319 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {62 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{11319 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(219\) |
default | \(\frac {\left (66 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-62 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-33 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+31 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-1860 x^{3}+884 x^{2}+3360 x +1296\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}}{1617 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(228\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=-\frac {360 \, {\left (31 \, x - 54\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1357 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 2790 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{145530 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
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\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=\int \frac {\sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {3 x + 2}}\, dx \]
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\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=\int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}} \,d x \]
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