Integrand size = 27, antiderivative size = 37 \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+c x}} \, dx=-\frac {2 E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {-b}}\right )\right |-1\right )}{\sqrt {-b} \sqrt {c}} \]
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Time = 0.02 (sec) , antiderivative size = 37, 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 = {111} \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+c x}} \, dx=-\frac {2 E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {b x}}{\sqrt {-b}}\right )\right |-1\right )}{\sqrt {-b} \sqrt {c}} \]
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Rule 111
Rubi steps \begin{align*} \text {integral}& = -\frac {2 E\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 = 0.72 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+c x}} \, dx=-\frac {2 x \left (-3 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^2 x^2\right )+c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^2 x^2\right )\right )}{3 \sqrt {b x}} \]
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Time = 0.62 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {2 \sqrt {2}\, \sqrt {-c x}\, E\left (\sqrt {c x +1}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {b x}}\) | \(33\) |
elliptic | \(\frac {\sqrt {-b x \left (c^{2} x^{2}-1\right )}\, \left (\frac {\sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {-2 c \left (x -\frac {1}{c}\right )}\, \sqrt {-c x}\, F\left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {-b \,c^{2} x^{3}+b x}}-\frac {\sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {-2 c \left (x -\frac {1}{c}\right )}\, \sqrt {-c x}\, \left (-\frac {2 E\left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {F\left (\sqrt {c \left (x +\frac {1}{c}\right )}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {-b \,c^{2} x^{3}+b x}}\right )}{\sqrt {b x}\, \sqrt {-c x +1}\, \sqrt {c x +1}}\) | \(183\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+c x}} \, dx=-\frac {2 \, {\left (\sqrt {-b c^{2}} c {\rm weierstrassZeta}\left (\frac {4}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right ) + \sqrt {-b c^{2}} {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right )}}{b c^{2}} \]
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\[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+c x}} \, dx=\int \frac {\sqrt {- c x + 1}}{\sqrt {b x} \sqrt {c x + 1}}\, dx \]
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\[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+c x}} \, dx=\int { \frac {\sqrt {-c x + 1}}{\sqrt {b x} \sqrt {c x + 1}} \,d x } \]
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\[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+c x}} \, dx=\int { \frac {\sqrt {-c x + 1}}{\sqrt {b x} \sqrt {c x + 1}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+c x}} \, dx=\int \frac {\sqrt {1-c\,x}}{\sqrt {b\,x}\,\sqrt {c\,x+1}} \,d x \]
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