Integrand size = 29, antiderivative size = 66 \[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{2} \left (\sqrt {x}+3 \sqrt {1+x}\right ) \sqrt {-x+\sqrt {x} \sqrt {1+x}}-\frac {3 \arcsin \left (\sqrt {x}-\sqrt {1+x}\right )}{2 \sqrt {2}} \]
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\[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx,x,\sqrt {1+x}\right ) \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\frac {1}{4} \left (2 \left (\sqrt {x}+3 \sqrt {1+x}\right ) \sqrt {-x+\sqrt {x} \sqrt {1+x}}-3 \sqrt {2} \arctan \left (\frac {\sqrt {-2 x+2 \sqrt {x} \sqrt {1+x}}}{-\sqrt {x}+\sqrt {1+x}}\right )\right ) \]
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\[\int \frac {\sqrt {-x +\sqrt {x}\, \sqrt {x +1}}}{\sqrt {x +1}}d x\]
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Timed out. \[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int \frac {\sqrt {\sqrt {x} \sqrt {x + 1} - x}}{\sqrt {x + 1}}\, dx \]
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\[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int { \frac {\sqrt {\sqrt {x + 1} \sqrt {x} - x}}{\sqrt {x + 1}} \,d x } \]
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\[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int { \frac {\sqrt {\sqrt {x + 1} \sqrt {x} - x}}{\sqrt {x + 1}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-x+\sqrt {x} \sqrt {1+x}}}{\sqrt {1+x}} \, dx=\int \frac {\sqrt {\sqrt {x}\,\sqrt {x+1}-x}}{\sqrt {x+1}} \,d x \]
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