Integrand size = 27, antiderivative size = 42 \[ \int \frac {1}{\sqrt {e x} \sqrt {2-b x} \sqrt {2+b x}} \, dx=\frac {\sqrt {2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {2} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt {e}} \]
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Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {117} \[ \int \frac {1}{\sqrt {e x} \sqrt {2-b x} \sqrt {2+b x}} \, dx=\frac {\sqrt {2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {2} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt {e}} \]
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Rule 117
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {2} \sqrt {e}}\right )\right |-1\right )}{\sqrt {b} \sqrt {e}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt {e x} \sqrt {2-b x} \sqrt {2+b x}} \, dx=\frac {x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b^2 x^2}{4}\right )}{\sqrt {e x}} \]
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Time = 2.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {F\left (\frac {\sqrt {b x +2}\, \sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {-b x}}{\sqrt {e x}\, b}\) | \(34\) |
elliptic | \(\frac {\sqrt {-e x \left (b^{2} x^{2}-4\right )}\, \sqrt {2}\, \sqrt {b \left (x +\frac {2}{b}\right )}\, \sqrt {-b \left (x -\frac {2}{b}\right )}\, \sqrt {-2 b x}\, F\left (\frac {\sqrt {2}\, \sqrt {b \left (x +\frac {2}{b}\right )}}{2}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {e x}\, \sqrt {-b x +2}\, \sqrt {b x +2}\, b \sqrt {-b^{2} e \,x^{3}+4 e x}}\) | \(111\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\sqrt {e x} \sqrt {2-b x} \sqrt {2+b x}} \, dx=-\frac {2 \, \sqrt {-b^{2} e} {\rm weierstrassPInverse}\left (\frac {16}{b^{2}}, 0, x\right )}{b^{2} e} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (39) = 78\).
Time = 11.50 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.50 \[ \int \frac {1}{\sqrt {e x} \sqrt {2-b x} \sqrt {2+b x}} \, dx=\frac {\sqrt {2} i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {4}{b^{2} x^{2}}} \right )}}{8 \pi ^{\frac {3}{2}} \sqrt {b} \sqrt {e}} - \frac {\sqrt {2} i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {4 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{8 \pi ^{\frac {3}{2}} \sqrt {b} \sqrt {e}} \]
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\[ \int \frac {1}{\sqrt {e x} \sqrt {2-b x} \sqrt {2+b x}} \, dx=\int { \frac {1}{\sqrt {b x + 2} \sqrt {-b x + 2} \sqrt {e x}} \,d x } \]
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\[ \int \frac {1}{\sqrt {e x} \sqrt {2-b x} \sqrt {2+b x}} \, dx=\int { \frac {1}{\sqrt {b x + 2} \sqrt {-b x + 2} \sqrt {e x}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {e x} \sqrt {2-b x} \sqrt {2+b x}} \, dx=\int \frac {1}{\sqrt {e\,x}\,\sqrt {2-b\,x}\,\sqrt {b\,x+2}} \,d x \]
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