Integrand size = 28, antiderivative size = 31 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=-\sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {114} \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=-\sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rule 114
Rubi steps \begin{align*} \text {integral}& = -\sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=i \sqrt {\frac {11}{3}} \left (E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(23)=46\).
Time = 1.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74
method | result | size |
default | \(\frac {\left (33 F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-35 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )\right ) \sqrt {-3-5 x}\, \sqrt {7}\, \sqrt {5}}{105 \sqrt {3+5 x}}\) | \(54\) |
elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {2 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{35 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{21 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(167\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=-\frac {31}{270} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {1}{3} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
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\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\int \frac {\sqrt {5 x + 3}}{\sqrt {1 - 2 x} \sqrt {3 x + 2}}\, dx \]
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\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{\sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}} \,d x } \]
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\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{\sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\int \frac {\sqrt {5\,x+3}}{\sqrt {1-2\,x}\,\sqrt {3\,x+2}} \,d x \]
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