Integrand size = 57, antiderivative size = 46 \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\sqrt {2} b \arcsin \left (\frac {a x-b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Time = 0.80 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2156, 2155, 222} \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\sqrt {2} b \arcsin \left (\frac {a x-b \sqrt {\frac {a^2 x^2}{b^2}+\frac {a}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \]
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Rule 222
Rule 2155
Rule 2156
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {-a x^2+b x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx \\ & = -\frac {\left (\sqrt {2} b\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}{a} \\ & = \frac {\sqrt {2} b \sin ^{-1}\left (\frac {a x-b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a}}\right )}{\sqrt {a}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(46)=92\).
Time = 0.01 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.48 \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\sqrt {2} b \sqrt {x \left (-a x+b \sqrt {\frac {a \left (1+a x^2\right )}{b^2}}\right )} \sqrt {a x \left (a x+b \sqrt {\frac {a \left (1+a x^2\right )}{b^2}}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {a x \left (a x+b \sqrt {\frac {a \left (1+a x^2\right )}{b^2}}\right )}}{\sqrt {a}}\right )}{a^{3/2} x} \]
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\[\int \frac {\sqrt {x \left (-a x +b \sqrt {\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\right )}}{x \sqrt {\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}d x\]
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none
Time = 5.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 3.50 \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\left [\frac {1}{2} \, \sqrt {2} b \sqrt {-\frac {1}{a}} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}} + 2 \, \sqrt {-a x^{2} + b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}} {\left (\sqrt {2} a x \sqrt {-\frac {1}{a}} - \sqrt {2} b \sqrt {-\frac {1}{a}} \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}\right )} + 1\right ), -\frac {\sqrt {2} b \arctan \left (\frac {\sqrt {2} \sqrt {-a x^{2} + b x \sqrt {\frac {a^{2} x^{2} + a}{b^{2}}}}}{2 \, \sqrt {a} x}\right )}{\sqrt {a}}\right ] \]
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\[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int \frac {\sqrt {- x \left (a x - b \sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}}\right )}}{x \sqrt {\frac {a \left (a x^{2} + 1\right )}{b^{2}}}}\, dx \]
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\[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int { \frac {\sqrt {-{\left (a x - \sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}} b\right )} x}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}} x} \,d x } \]
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\[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int { \frac {\sqrt {-{\left (a x - \sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}} b\right )} x}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} + \frac {a}{b^{2}}} x} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {x \left (-a x+b \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )}}{x \sqrt {\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int \frac {\sqrt {-x\,\left (a\,x-b\,\sqrt {\frac {a}{b^2}+\frac {a^2\,x^2}{b^2}}\right )}}{x\,\sqrt {\frac {a}{b^2}+\frac {a^2\,x^2}{b^2}}} \,d x \]
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