Integrand size = 27, antiderivative size = 42 \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=-\frac {2 E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {b x}}{\sqrt {-b}}\right )|-\frac {c}{d}\right )}{\sqrt {-b} \sqrt {d}} \]
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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 = {111} \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=-\frac {2 E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {b x}}{\sqrt {-b}}\right )|-\frac {c}{d}\right )}{\sqrt {-b} \sqrt {d}} \]
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Rule 111
Rubi steps \begin{align*} \text {integral}& = -\frac {2 E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {b x}}{\sqrt {-b}}\right )|-\frac {c}{d}\right )}{\sqrt {-b} \sqrt {d}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(112\) vs. \(2(42)=84\).
Time = 3.90 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.67 \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=\frac {-\frac {2 \sqrt {\frac {1}{c}} (-1+c x) (1+d x)}{d}-2 \sqrt {1-\frac {1}{c x}} \sqrt {1+\frac {1}{d x}} x^{3/2} E\left (\arcsin \left (\frac {\sqrt {\frac {1}{c}}}{\sqrt {x}}\right )|-\frac {c}{d}\right )}{\sqrt {\frac {1}{c}} \sqrt {b x} \sqrt {1-c x} \sqrt {1+d x}} \]
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Time = 1.79 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.60
method | result | size |
default | \(-\frac {2 \left (c +d \right ) E\left (\sqrt {d x +1}, \sqrt {\frac {c}{c +d}}\right ) \sqrt {-d x}\, \sqrt {-\frac {\left (c x -1\right ) d}{c +d}}\, \sqrt {-c x +1}}{\left (c x -1\right ) \sqrt {b x}\, d^{2}}\) | \(67\) |
elliptic | \(\frac {\sqrt {-b x \left (c x -1\right ) \left (d x +1\right )}\, \left (\frac {2 \sqrt {\left (x +\frac {1}{d}\right ) d}\, \sqrt {\frac {x -\frac {1}{c}}{-\frac {1}{d}-\frac {1}{c}}}\, \sqrt {-d x}\, F\left (\sqrt {\left (x +\frac {1}{d}\right ) d}, \sqrt {-\frac {1}{d \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )}{d \sqrt {-b c d \,x^{3}-c b \,x^{2}+b d \,x^{2}+b x}}-\frac {2 c \sqrt {\left (x +\frac {1}{d}\right ) d}\, \sqrt {\frac {x -\frac {1}{c}}{-\frac {1}{d}-\frac {1}{c}}}\, \sqrt {-d x}\, \left (\left (-\frac {1}{d}-\frac {1}{c}\right ) E\left (\sqrt {\left (x +\frac {1}{d}\right ) d}, \sqrt {-\frac {1}{d \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )+\frac {F\left (\sqrt {\left (x +\frac {1}{d}\right ) d}, \sqrt {-\frac {1}{d \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )}{c}\right )}{d \sqrt {-b c d \,x^{3}-c b \,x^{2}+b d \,x^{2}+b x}}\right )}{\sqrt {b x}\, \sqrt {-c x +1}\, \sqrt {d x +1}}\) | \(287\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 222, normalized size of antiderivative = 5.29 \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {-b c d} c d {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, -\frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, -\frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, \frac {3 \, c d x + c - d}{3 \, c d}\right )\right ) + \sqrt {-b c d} {\left (c + 2 \, d\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, -\frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, \frac {3 \, c d x + c - d}{3 \, c d}\right )\right )}}{3 \, b c d^{2}} \]
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\[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=\int \frac {\sqrt {- c x + 1}}{\sqrt {b x} \sqrt {d x + 1}}\, dx \]
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\[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=\int { \frac {\sqrt {-c x + 1}}{\sqrt {b x} \sqrt {d x + 1}} \,d x } \]
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\[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=\int { \frac {\sqrt {-c x + 1}}{\sqrt {b x} \sqrt {d x + 1}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1-c x}}{\sqrt {b x} \sqrt {1+d x}} \, dx=\int \frac {\sqrt {1-c\,x}}{\sqrt {b\,x}\,\sqrt {d\,x+1}} \,d x \]
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