Integrand size = 27, antiderivative size = 33 \[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+c x}} \, dx=\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {b} \sqrt {c}} \]
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Time = 0.01 (sec) , antiderivative size = 33, 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} \[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+c x}} \, dx=\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {b} \sqrt {c}} \]
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Rule 117
Rubi steps \begin{align*} \text {integral}& = \frac {2 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {b} \sqrt {c}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+c x}} \, dx=\frac {2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right )}{\sqrt {b x}} \]
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Time = 0.66 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {-c x}\, F\left (\sqrt {c x +1}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {b x}}\) | \(32\) |
elliptic | \(\frac {\sqrt {-b x \left (c^{2} x^{2}-1\right )}\, \sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {-2 c \left (x -\frac {1}{c}\right )}\, \sqrt {-c x}\, F\left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b x}\, \sqrt {-c x +1}\, \sqrt {c x +1}\, c \sqrt {-b \,c^{2} x^{3}+b x}}\) | \(97\) |
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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 = 0.73 \[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+c x}} \, dx=-\frac {2 \, \sqrt {-b c^{2}} {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )}{b c^{2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (29) = 58\).
Time = 11.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.85 \[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+c x}} \, dx=\frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {b} \sqrt {c}} - \frac {i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {b} \sqrt {c}} \]
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\[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+c x}} \, dx=\int { \frac {1}{\sqrt {b x} \sqrt {c x + 1} \sqrt {-c x + 1}} \,d x } \]
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\[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+c x}} \, dx=\int { \frac {1}{\sqrt {b x} \sqrt {c x + 1} \sqrt {-c x + 1}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {b x} \sqrt {1-c x} \sqrt {1+c x}} \, dx=\int \frac {1}{\sqrt {b\,x}\,\sqrt {1-c\,x}\,\sqrt {c\,x+1}} \,d x \]
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