Integrand size = 27, antiderivative size = 38 \[ \int \frac {\sqrt {1+c x}}{\sqrt {b x} \sqrt {1-d x}} \, dx=\frac {2 E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {b x}}{\sqrt {b}}\right )|-\frac {c}{d}\right )}{\sqrt {b} \sqrt {d}} \]
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Time = 0.01 (sec) , antiderivative size = 38, 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. \(102\) vs. \(2(38)=76\).
Time = 3.79 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.68 \[ \int \frac {\sqrt {1+c x}}{\sqrt {b x} \sqrt {1-d x}} \, dx=\frac {2 \sqrt {1-d x} \left (-1-c x+\frac {\sqrt {1+\frac {1}{c x}} \sqrt {x} E\left (\arcsin \left (\frac {\sqrt {-\frac {1}{c}}}{\sqrt {x}}\right )|-\frac {c}{d}\right )}{\sqrt {-\frac {1}{c}} \sqrt {1-\frac {1}{d x}}}\right )}{d \sqrt {b x} \sqrt {1+c x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(128\) vs. \(2(29)=58\).
Time = 2.35 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.39
method | result | size |
default | \(-\frac {2 \left (F\left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right ) c +F\left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right ) d -E\left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right ) c -E\left (\sqrt {c x +1}, \sqrt {\frac {d}{c +d}}\right ) d \right ) \sqrt {-c x}\, \sqrt {-\frac {\left (d x -1\right ) c}{c +d}}\, \sqrt {-d x +1}}{\left (d x -1\right ) \sqrt {b x}\, c d}\) | \(129\) |
elliptic | \(\frac {\sqrt {-b x \left (d x -1\right ) \left (c x +1\right )}\, \left (\frac {2 \sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {\frac {x -\frac {1}{d}}{-\frac {1}{d}-\frac {1}{c}}}\, \sqrt {-c x}\, F\left (\sqrt {c \left (x +\frac {1}{c}\right )}, \sqrt {-\frac {1}{c \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )}{c \sqrt {-b c d \,x^{3}+c b \,x^{2}-b d \,x^{2}+b x}}+\frac {2 \sqrt {c \left (x +\frac {1}{c}\right )}\, \sqrt {\frac {x -\frac {1}{d}}{-\frac {1}{d}-\frac {1}{c}}}\, \sqrt {-c x}\, \left (\left (-\frac {1}{d}-\frac {1}{c}\right ) E\left (\sqrt {c \left (x +\frac {1}{c}\right )}, \sqrt {-\frac {1}{c \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )+\frac {F\left (\sqrt {c \left (x +\frac {1}{c}\right )}, \sqrt {-\frac {1}{c \left (-\frac {1}{d}-\frac {1}{c}\right )}}\right )}{d}\right )}{\sqrt {-b c d \,x^{3}+c b \,x^{2}-b d \,x^{2}+b x}}\right )}{\sqrt {b x}\, \sqrt {c x +1}\, \sqrt {-d x +1}}\) | \(283\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 223, normalized size of antiderivative = 5.87 \[ \int \frac {\sqrt {1+c x}}{\sqrt {b x} \sqrt {1-d x}} \, dx=\frac {2 \, {\left (3 \, \sqrt {-b c d} c d {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, \frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, \frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, \frac {3 \, c d x - c + d}{3 \, c d}\right )\right ) - \sqrt {-b c d} {\left (c + 2 \, d\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} + c d + d^{2}\right )}}{3 \, c^{2} d^{2}}, \frac {4 \, {\left (2 \, c^{3} + 3 \, c^{2} d - 3 \, c d^{2} - 2 \, d^{3}\right )}}{27 \, c^{3} d^{3}}, \frac {3 \, c d x - c + d}{3 \, c d}\right )\right )}}{3 \, b c d^{2}} \]
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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. \[ \int \frac {\sqrt {1+c x}}{\sqrt {b x} \sqrt {1-d x}} \, dx=\int \frac {\sqrt {c\,x+1}}{\sqrt {b\,x}\,\sqrt {1-d\,x}} \,d x \]
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