Integrand size = 30, antiderivative size = 130 \[ \int \frac {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {\left (255-136 x-384 x^2\right ) \sqrt {x+x^2} \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{960 x}+\sqrt {x \left (x+\sqrt {x+x^2}\right )} \left (\frac {1}{960} \left (-85+568 x+384 x^2\right )-\frac {17 \sqrt {-x+\sqrt {x+x^2}} \text {arctanh}\left (\sqrt {2} \sqrt {-x+\sqrt {x+x^2}}\right )}{64 \sqrt {2} x}\right ) \]
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\[ \int \frac {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int \frac {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x+x^2} \int \frac {x^{3/2} \sqrt {1+x}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx}{\sqrt {x} \sqrt {1+x}} \\ & = \frac {\left (2 \sqrt {x+x^2}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt {1+x^2}}{\sqrt {x^4+x^2 \sqrt {x^2+x^4}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x}} \\ \end{align*}
Time = 2.69 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.09 \[ \int \frac {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {(1+x) \sqrt {x \left (x+\sqrt {x (1+x)}\right )} \left (2 x \left (-384 x^3-85 \left (-3+\sqrt {x (1+x)}\right )+8 x^2 \left (-65+48 \sqrt {x (1+x)}\right )+x \left (119+568 \sqrt {x (1+x)}\right )\right )-255 \sqrt {2} \sqrt {x (1+x)} \sqrt {-x+\sqrt {x (1+x)}} \text {arctanh}\left (\sqrt {-2 x+2 \sqrt {x (1+x)}}\right )\right )}{1920 (x (1+x))^{3/2}} \]
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\[\int \frac {x \sqrt {x^{2}+x}}{\sqrt {x^{2}+x \sqrt {x^{2}+x}}}d x\]
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Time = 0.24 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91 \[ \int \frac {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {255 \, \sqrt {2} x \log \left (\frac {4 \, x^{2} - 2 \, \sqrt {x^{2} + \sqrt {x^{2} + x} x} {\left (\sqrt {2} x + \sqrt {2} \sqrt {x^{2} + x}\right )} + 4 \, \sqrt {x^{2} + x} x + x}{x}\right ) + 4 \, {\left (384 \, x^{3} + 568 \, x^{2} - {\left (384 \, x^{2} + 136 \, x - 255\right )} \sqrt {x^{2} + x} - 85 \, x\right )} \sqrt {x^{2} + \sqrt {x^{2} + x} x}}{3840 \, x} \]
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\[ \int \frac {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int \frac {x \sqrt {x \left (x + 1\right )}}{\sqrt {x \left (x + \sqrt {x^{2} + x}\right )}}\, dx \]
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\[ \int \frac {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int { \frac {\sqrt {x^{2} + x} x}{\sqrt {x^{2} + \sqrt {x^{2} + x} x}} \,d x } \]
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\[ \int \frac {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int { \frac {\sqrt {x^{2} + x} x}{\sqrt {x^{2} + \sqrt {x^{2} + x} x}} \,d x } \]
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Timed out. \[ \int \frac {x \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\int \frac {x\,\sqrt {x^2+x}}{\sqrt {x^2+x\,\sqrt {x^2+x}}} \,d x \]
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