Integrand size = 28, antiderivative size = 81 \[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+\frac {2 \sqrt {\frac {7}{5}} \sqrt {-3-5 x} E\left (\arcsin \left (\sqrt {5} \sqrt {2+3 x}\right )|\frac {2}{35}\right )}{11 \sqrt {3+5 x}} \]
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Time = 0.02 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {101, 21, 115, 114} \[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {\frac {7}{5}} \sqrt {-5 x-3} E\left (\arcsin \left (\sqrt {5} \sqrt {3 x+2}\right )|\frac {2}{35}\right )}{11 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}} \]
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Rule 21
Rule 101
Rule 114
Rule 115
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+\frac {2}{11} \int \frac {\frac {3}{2}-3 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+\frac {3}{11} \int \frac {\sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx \\ & = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+\frac {\left (3 \sqrt {7} \sqrt {-3-5 x}\right ) \int \frac {\sqrt {\frac {3}{7}-\frac {6 x}{7}}}{\sqrt {-9-15 x} \sqrt {2+3 x}} \, dx}{11 \sqrt {3+5 x}} \\ & = -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+\frac {2 \sqrt {\frac {7}{5}} \sqrt {-3-5 x} E\left (\sin ^{-1}\left (\sqrt {5} \sqrt {2+3 x}\right )|\frac {2}{35}\right )}{11 \sqrt {3+5 x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.62 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\frac {2}{55} \left (-\frac {5 \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}}-i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )\right ) \]
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Time = 1.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.14
method | result | size |
default | \(-\frac {2 \sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (\sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+30 x^{2}+5 x -10\right )}{55 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(92\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-30 x^{2}-5 x +10\right )}{55 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {2 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{385 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{385 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(195\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (34 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 45 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) + 225 \, \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}\right )}}{2475 \, {\left (5 \, x + 3\right )}} \]
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\[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\int \frac {\sqrt {3 x + 2}}{\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
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\[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\int { \frac {\sqrt {3 \, x + 2}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\int \frac {\sqrt {3\,x+2}}{\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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