Integrand size = 37, antiderivative size = 121 \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=-\frac {4 \sqrt {-x+x^2} \sqrt {x \left (x-\sqrt {-x+x^2}\right )}}{3 x^2}+\sqrt {x \left (x-\sqrt {-x+x^2}\right )} \left (\frac {4}{3 x}-\frac {2 \sqrt {2} \sqrt {x+\sqrt {-x+x^2}} \text {arctanh}\left (\sqrt {2} \sqrt {x+\sqrt {-x+x^2}}\right )}{x}\right ) \]
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\[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-x+x^2} \int \frac {\sqrt {-1+x} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^{5/2}} \, dx}{\sqrt {-1+x} \sqrt {x}} \\ & = \frac {\left (2 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2} \sqrt {x^4-x^2 \sqrt {-x^2+x^4}}}{x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {-1+x} \sqrt {x}} \\ \end{align*}
Time = 2.52 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=-\frac {2 \sqrt {x \left (x-\sqrt {(-1+x) x}\right )} \left (2 x-2 \left (1+\sqrt {(-1+x) x}\right )+3 \sqrt {2} \sqrt {(-1+x) x} \sqrt {x+\sqrt {(-1+x) x}} \text {arctanh}\left (\sqrt {2} \sqrt {x+\sqrt {(-1+x) x}}\right )\right )}{3 x \sqrt {(-1+x) x}} \]
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\[\int \frac {\sqrt {x^{2}-x}\, \sqrt {x^{2}-x \sqrt {x^{2}-x}}}{x^{3}}d x\]
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Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\frac {3 \, \sqrt {2} x^{2} \log \left (-\frac {4 \, x^{2} - 2 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (\sqrt {2} x - \sqrt {2} \sqrt {x^{2} - x}\right )} - 4 \, \sqrt {x^{2} - x} x - x}{x}\right ) + 4 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (x - \sqrt {x^{2} - x}\right )}}{3 \, x^{2}} \]
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\[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int \frac {\sqrt {x \left (x - 1\right )} \sqrt {x \left (x - \sqrt {x^{2} - x}\right )}}{x^{3}}\, dx \]
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\[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int { \frac {\sqrt {x^{2} - \sqrt {x^{2} - x} x} \sqrt {x^{2} - x}}{x^{3}} \,d x } \]
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\[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int { \frac {\sqrt {x^{2} - \sqrt {x^{2} - x} x} \sqrt {x^{2} - x}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx=\int \frac {\sqrt {x^2-x}\,\sqrt {x^2-x\,\sqrt {x^2-x}}}{x^3} \,d x \]
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