Integrand size = 29, antiderivative size = 35 \[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1+a x}} \, dx=\sqrt {x} \sqrt {1-a x}+\frac {\arcsin \left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \]
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Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {862, 52, 56, 222} \[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1+a x}} \, dx=\frac {\arcsin \left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}+\sqrt {x} \sqrt {1-a x} \]
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Rule 52
Rule 56
Rule 222
Rule 862
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1-a x}}{\sqrt {x}} \, dx \\ & = \sqrt {x} \sqrt {1-a x}+\frac {1}{2} \int \frac {1}{\sqrt {x} \sqrt {1-a x}} \, dx \\ & = \sqrt {x} \sqrt {1-a x}+\text {Subst}\left (\int \frac {1}{\sqrt {1-a x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \sqrt {x} \sqrt {1-a x}+\frac {\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \\ \end{align*}
Time = 1.52 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1+a x}} \, dx=\sqrt {x} \sqrt {1-a x}+\frac {\arcsin \left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(25)=50\).
Time = 0.39 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.17
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {x}\, \left (2 \sqrt {a}\, \sqrt {-x \left (a x -1\right )}+\arctan \left (\frac {2 a x -1}{2 \sqrt {a}\, \sqrt {-x \left (a x -1\right )}}\right )\right )}{2 \sqrt {a x +1}\, \sqrt {-x \left (a x -1\right )}\, \sqrt {a}}\) | \(76\) |
risch | \(-\frac {\sqrt {x}\, \left (a x -1\right ) \sqrt {\frac {x \left (-a^{2} x^{2}+1\right )}{a x +1}}\, \sqrt {a x +1}}{\sqrt {-x \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a \,x^{2}+x}}\right ) \sqrt {\frac {x \left (-a^{2} x^{2}+1\right )}{a x +1}}\, \sqrt {a x +1}}{2 \sqrt {a}\, \sqrt {x}\, \sqrt {-a^{2} x^{2}+1}}\) | \(132\) |
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (25) = 50\).
Time = 0.31 (sec) , antiderivative size = 199, normalized size of antiderivative = 5.69 \[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1+a x}} \, dx=\left [\frac {4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {a x + 1} a \sqrt {x} - {\left (a x + 1\right )} \sqrt {-a} \log \left (-\frac {8 \, a^{3} x^{3} - 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x - 1\right )} \sqrt {a x + 1} \sqrt {-a} \sqrt {x} - 7 \, a x + 1}{a x + 1}\right )}{4 \, {\left (a^{2} x + a\right )}}, \frac {2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {a x + 1} a \sqrt {x} - {\left (a x + 1\right )} \sqrt {a} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {a x + 1} \sqrt {a} \sqrt {x}}{2 \, a^{2} x^{2} + a x - 1}\right )}{2 \, {\left (a^{2} x + a\right )}}\right ] \]
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\[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1+a x}} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {x} \sqrt {a x + 1}}\, dx \]
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\[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1+a x}} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{\sqrt {a x + 1} \sqrt {x}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (25) = 50\).
Time = 5.44 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.71 \[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1+a x}} \, dx=\frac {a {\left (\frac {\sqrt {2} - \log \left ({\left | -\sqrt {2} \sqrt {-a} + \sqrt {-a} \right |}\right )}{\sqrt {-a}} + \frac {\log \left ({\left | -\sqrt {-a x + 1} \sqrt {-a} + \sqrt {{\left (a x - 1\right )} a + a} \right |}\right )}{\sqrt {-a}} + \frac {\sqrt {{\left (a x - 1\right )} a + a} \sqrt {-a x + 1}}{a}\right )}}{{\left | a \right |}} \]
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Timed out. \[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1+a x}} \, dx=\int \frac {\sqrt {1-a^2\,x^2}}{\sqrt {x}\,\sqrt {a\,x+1}} \,d x \]
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