Integrand size = 23, antiderivative size = 158 \[ \int \sqrt {x} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\frac {\sqrt {x}}{12}-\frac {2 x^{3/2}}{9}-\frac {11 \sqrt {-x+x^2}}{12 \sqrt {x}}-\frac {2 \left (-x+x^2\right )^{3/2}}{9 x^{3/2}}+\frac {\sqrt {-x+x^2} \arctan \left (\frac {2}{3} \sqrt {2} \sqrt {-1+x}\right )}{24 \sqrt {2} \sqrt {-1+x} \sqrt {x}}-\frac {\arctan \left (2 \sqrt {2} \sqrt {x}\right )}{24 \sqrt {2}}+\frac {2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2617, 2615, 6865, 6874, 209, 1602, 2025, 1160, 455, 52, 65, 210} \[ \int \sqrt {x} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\frac {\sqrt {x^2-x} \arctan \left (\frac {2}{3} \sqrt {2} \sqrt {x-1}\right )}{24 \sqrt {2} \sqrt {x-1} \sqrt {x}}-\frac {\arctan \left (2 \sqrt {2} \sqrt {x}\right )}{24 \sqrt {2}}-\frac {2 x^{3/2}}{9}-\frac {11 \sqrt {x^2-x}}{12 \sqrt {x}}-\frac {2 \left (x^2-x\right )^{3/2}}{9 x^{3/2}}+\frac {2}{3} x^{3/2} \log \left (4 \sqrt {x^2-x}+4 x-1\right )+\frac {\sqrt {x}}{12} \]
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Rule 52
Rule 65
Rule 209
Rule 210
Rule 455
Rule 1160
Rule 1602
Rule 2025
Rule 2615
Rule 2617
Rule 6865
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {x} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right ) \, dx \\ & = \frac {2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {16}{3} \int \frac {x^{3/2}}{-4 (1+2 x) \sqrt {-x+x^2}+8 \left (-x+x^2\right )} \, dx \\ & = \frac {2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {32}{3} \text {Subst}\left (\int \frac {x^4}{-4 \left (1+2 x^2\right ) \sqrt {-x^2+x^4}+8 \left (-x^2+x^4\right )} \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {32}{3} \text {Subst}\left (\int \left (\frac {1}{128}-\frac {x^2}{16}-\frac {1}{128 \left (1+8 x^2\right )}-\frac {x^2}{12 \sqrt {-x^2+x^4}}-\frac {1}{16} \sqrt {-x^2+x^4}+\frac {\sqrt {-x^2+x^4}}{48 \left (-1-8 x^2\right )}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {\sqrt {x}}{12}-\frac {2 x^{3/2}}{9}+\frac {2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )-\frac {1}{12} \text {Subst}\left (\int \frac {1}{1+8 x^2} \, dx,x,\sqrt {x}\right )+\frac {2}{9} \text {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{-1-8 x^2} \, dx,x,\sqrt {x}\right )-\frac {2}{3} \text {Subst}\left (\int \sqrt {-x^2+x^4} \, dx,x,\sqrt {x}\right )-\frac {8}{9} \text {Subst}\left (\int \frac {x^2}{\sqrt {-x^2+x^4}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {\sqrt {x}}{12}-\frac {2 x^{3/2}}{9}-\frac {8 \sqrt {-x+x^2}}{9 \sqrt {x}}-\frac {2 \left (-x+x^2\right )^{3/2}}{9 x^{3/2}}-\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{24 \sqrt {2}}+\frac {2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {\left (2 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {x \sqrt {-1+x^2}}{-1-8 x^2} \, dx,x,\sqrt {x}\right )}{9 \sqrt {-1+x} \sqrt {x}} \\ & = \frac {\sqrt {x}}{12}-\frac {2 x^{3/2}}{9}-\frac {8 \sqrt {-x+x^2}}{9 \sqrt {x}}-\frac {2 \left (-x+x^2\right )^{3/2}}{9 x^{3/2}}-\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{24 \sqrt {2}}+\frac {2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {\sqrt {-x+x^2} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{-1-8 x} \, dx,x,x\right )}{9 \sqrt {-1+x} \sqrt {x}} \\ & = \frac {\sqrt {x}}{12}-\frac {2 x^{3/2}}{9}-\frac {11 \sqrt {-x+x^2}}{12 \sqrt {x}}-\frac {2 \left (-x+x^2\right )^{3/2}}{9 x^{3/2}}-\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{24 \sqrt {2}}+\frac {2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )-\frac {\sqrt {-x+x^2} \text {Subst}\left (\int \frac {1}{(-1-8 x) \sqrt {-1+x}} \, dx,x,x\right )}{8 \sqrt {-1+x} \sqrt {x}} \\ & = \frac {\sqrt {x}}{12}-\frac {2 x^{3/2}}{9}-\frac {11 \sqrt {-x+x^2}}{12 \sqrt {x}}-\frac {2 \left (-x+x^2\right )^{3/2}}{9 x^{3/2}}-\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{24 \sqrt {2}}+\frac {2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )-\frac {\sqrt {-x+x^2} \text {Subst}\left (\int \frac {1}{-9-8 x^2} \, dx,x,\sqrt {-1+x}\right )}{4 \sqrt {-1+x} \sqrt {x}} \\ & = \frac {\sqrt {x}}{12}-\frac {2 x^{3/2}}{9}-\frac {11 \sqrt {-x+x^2}}{12 \sqrt {x}}-\frac {2 \left (-x+x^2\right )^{3/2}}{9 x^{3/2}}+\frac {\sqrt {-x+x^2} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {-1+x}\right )}{24 \sqrt {2} \sqrt {-1+x} \sqrt {x}}-\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{24 \sqrt {2}}+\frac {2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.25 \[ \int \sqrt {x} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\frac {-3 \sqrt {2} \sqrt {(-1+x) x} \arctan \left (\frac {2 \sqrt {2}-i \sqrt {x}}{3 \sqrt {-1+x}}\right )-3 \sqrt {2} \sqrt {(-1+x) x} \arctan \left (\frac {2 \sqrt {2}+i \sqrt {x}}{3 \sqrt {-1+x}}\right )+2 \sqrt {-1+x} \left (-3 \sqrt {2} \sqrt {x} \arctan \left (2 \sqrt {2} \sqrt {x}\right )-4 \left (-3 x+8 x^2+25 \sqrt {(-1+x) x}+8 x \sqrt {(-1+x) x}-24 x^2 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )\right )\right )}{288 \sqrt {-1+x} \sqrt {x}} \]
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\[\int \sqrt {x}\, \ln \left (-1+4 x +4 \sqrt {\left (-1+x \right ) x}\right )d x\]
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none
Time = 0.33 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.63 \[ \int \sqrt {x} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\frac {96 \, x^{\frac {5}{2}} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) - 3 \, \sqrt {2} x \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) - 3 \, \sqrt {2} x \arctan \left (\frac {3 \, \sqrt {2} \sqrt {x}}{4 \, \sqrt {x^{2} - x}}\right ) - 4 \, \sqrt {x^{2} - x} {\left (8 \, x + 25\right )} \sqrt {x} - 4 \, {\left (8 \, x^{2} - 3 \, x\right )} \sqrt {x}}{144 \, x} \]
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Timed out. \[ \int \sqrt {x} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\text {Timed out} \]
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\[ \int \sqrt {x} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\int { \sqrt {x} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right ) \,d x } \]
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Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77 \[ \int \sqrt {x} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\frac {2}{3} \, x^{\frac {3}{2}} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right ) - \frac {1}{96} \, \sqrt {2} {\left (\pi - 2 \, \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1\right )}}{3 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right )\right )} + \frac {1}{288} \, \sqrt {2} {\left (-3 i \, \pi + 100 i \, \sqrt {2} - 6 \, \arctan \left (\frac {2}{3} i \, \sqrt {2}\right )\right )} - \frac {1}{36} \, {\left (8 \, x + 25\right )} \sqrt {x - 1} - \frac {2}{9} \, x^{\frac {3}{2}} - \frac {1}{48} \, \sqrt {2} \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) + \frac {1}{12} \, \sqrt {x} \]
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Timed out. \[ \int \sqrt {x} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\int \sqrt {x}\,\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right ) \,d x \]
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