Integrand size = 23, antiderivative size = 187 \[ \int x^{3/2} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=-\frac {\sqrt {x}}{160}+\frac {x^{3/2}}{60}-\frac {2 x^{5/2}}{25}-\frac {17 \sqrt {-x+x^2}}{32 \sqrt {x}}-\frac {71 \left (-x+x^2\right )^{3/2}}{300 x^{3/2}}-\frac {2 \left (-x+x^2\right )^{3/2}}{25 \sqrt {x}}-\frac {\sqrt {-x+x^2} \arctan \left (\frac {2}{3} \sqrt {2} \sqrt {-1+x}\right )}{320 \sqrt {2} \sqrt {-1+x} \sqrt {x}}+\frac {\arctan \left (2 \sqrt {2} \sqrt {x}\right )}{320 \sqrt {2}}+\frac {2}{5} x^{5/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right ) \]
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Time = 0.36 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2617, 2615, 6865, 6874, 209, 1602, 2025, 2041, 1160, 455, 52, 65} \[ \int x^{3/2} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=-\frac {\sqrt {x^2-x} \arctan \left (\frac {2}{3} \sqrt {2} \sqrt {x-1}\right )}{320 \sqrt {2} \sqrt {x-1} \sqrt {x}}+\frac {\arctan \left (2 \sqrt {2} \sqrt {x}\right )}{320 \sqrt {2}}-\frac {2 x^{5/2}}{25}+\frac {x^{3/2}}{60}-\frac {2 \left (x^2-x\right )^{3/2}}{25 \sqrt {x}}-\frac {17 \sqrt {x^2-x}}{32 \sqrt {x}}-\frac {71 \left (x^2-x\right )^{3/2}}{300 x^{3/2}}+\frac {2}{5} x^{5/2} \log \left (4 \sqrt {x^2-x}+4 x-1\right )-\frac {\sqrt {x}}{160} \]
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Rule 52
Rule 65
Rule 209
Rule 455
Rule 1160
Rule 1602
Rule 2025
Rule 2041
Rule 2615
Rule 2617
Rule 6865
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int x^{3/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right ) \, dx \\ & = \frac {2}{5} x^{5/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {16}{5} \int \frac {x^{5/2}}{-4 (1+2 x) \sqrt {-x+x^2}+8 \left (-x+x^2\right )} \, dx \\ & = \frac {2}{5} x^{5/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {32}{5} \text {Subst}\left (\int \frac {x^6}{-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}{5} x^{5/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {32}{5} \text {Subst}\left (\int \left (-\frac {1}{1024}+\frac {x^2}{128}-\frac {x^4}{16}+\frac {1}{1024 \left (1+8 x^2\right )}-\frac {x^2}{12 \sqrt {-x^2+x^4}}-\frac {11}{128} \sqrt {-x^2+x^4}-\frac {1}{16} x^2 \sqrt {-x^2+x^4}+\frac {\sqrt {-x^2+x^4}}{384 \left (1+8 x^2\right )}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\sqrt {x}}{160}+\frac {x^{3/2}}{60}-\frac {2 x^{5/2}}{25}+\frac {2}{5} x^{5/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {1}{160} \text {Subst}\left (\int \frac {1}{1+8 x^2} \, dx,x,\sqrt {x}\right )+\frac {1}{60} \text {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{1+8 x^2} \, dx,x,\sqrt {x}\right )-\frac {2}{5} \text {Subst}\left (\int x^2 \sqrt {-x^2+x^4} \, dx,x,\sqrt {x}\right )-\frac {8}{15} \text {Subst}\left (\int \frac {x^2}{\sqrt {-x^2+x^4}} \, dx,x,\sqrt {x}\right )-\frac {11}{20} \text {Subst}\left (\int \sqrt {-x^2+x^4} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {\sqrt {x}}{160}+\frac {x^{3/2}}{60}-\frac {2 x^{5/2}}{25}-\frac {8 \sqrt {-x+x^2}}{15 \sqrt {x}}-\frac {11 \left (-x+x^2\right )^{3/2}}{60 x^{3/2}}-\frac {2 \left (-x+x^2\right )^{3/2}}{25 \sqrt {x}}+\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{320 \sqrt {2}}+\frac {2}{5} x^{5/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )-\frac {4}{25} \text {Subst}\left (\int \sqrt {-x^2+x^4} \, dx,x,\sqrt {x}\right )+\frac {\sqrt {-x+x^2} \text {Subst}\left (\int \frac {x \sqrt {-1+x^2}}{1+8 x^2} \, dx,x,\sqrt {x}\right )}{60 \sqrt {-1+x} \sqrt {x}} \\ & = -\frac {\sqrt {x}}{160}+\frac {x^{3/2}}{60}-\frac {2 x^{5/2}}{25}-\frac {8 \sqrt {-x+x^2}}{15 \sqrt {x}}-\frac {71 \left (-x+x^2\right )^{3/2}}{300 x^{3/2}}-\frac {2 \left (-x+x^2\right )^{3/2}}{25 \sqrt {x}}+\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{320 \sqrt {2}}+\frac {2}{5} x^{5/2} \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 )}{120 \sqrt {-1+x} \sqrt {x}} \\ & = -\frac {\sqrt {x}}{160}+\frac {x^{3/2}}{60}-\frac {2 x^{5/2}}{25}-\frac {17 \sqrt {-x+x^2}}{32 \sqrt {x}}-\frac {71 \left (-x+x^2\right )^{3/2}}{300 x^{3/2}}-\frac {2 \left (-x+x^2\right )^{3/2}}{25 \sqrt {x}}+\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{320 \sqrt {2}}+\frac {2}{5} x^{5/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )-\frac {\left (3 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} (1+8 x)} \, dx,x,x\right )}{320 \sqrt {-1+x} \sqrt {x}} \\ & = -\frac {\sqrt {x}}{160}+\frac {x^{3/2}}{60}-\frac {2 x^{5/2}}{25}-\frac {17 \sqrt {-x+x^2}}{32 \sqrt {x}}-\frac {71 \left (-x+x^2\right )^{3/2}}{300 x^{3/2}}-\frac {2 \left (-x+x^2\right )^{3/2}}{25 \sqrt {x}}+\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{320 \sqrt {2}}+\frac {2}{5} x^{5/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )-\frac {\left (3 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{9+8 x^2} \, dx,x,\sqrt {-1+x}\right )}{160 \sqrt {-1+x} \sqrt {x}} \\ & = -\frac {\sqrt {x}}{160}+\frac {x^{3/2}}{60}-\frac {2 x^{5/2}}{25}-\frac {17 \sqrt {-x+x^2}}{32 \sqrt {x}}-\frac {71 \left (-x+x^2\right )^{3/2}}{300 x^{3/2}}-\frac {2 \left (-x+x^2\right )^{3/2}}{25 \sqrt {x}}-\frac {\sqrt {-x+x^2} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {-1+x}\right )}{320 \sqrt {2} \sqrt {-1+x} \sqrt {x}}+\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{320 \sqrt {2}}+\frac {2}{5} x^{5/2} \log \left (-1+4 x+4 \sqrt {-x+x^2}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.93 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.16 \[ \int x^{3/2} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\frac {15 \sqrt {2} \sqrt {(-1+x) x} \arctan \left (\frac {2 \sqrt {2}-i \sqrt {x}}{3 \sqrt {-1+x}}\right )+15 \sqrt {2} \sqrt {(-1+x) x} \arctan \left (\frac {2 \sqrt {2}+i \sqrt {x}}{3 \sqrt {-1+x}}\right )-2 \sqrt {-1+x} \left (-15 \sqrt {2} \sqrt {x} \arctan \left (2 \sqrt {2} \sqrt {x}\right )+4 \left (192 x^3+707 \sqrt {(-1+x) x}+8 x^2 \left (-5+24 \sqrt {(-1+x) x}\right )+x \left (15+376 \sqrt {(-1+x) x}\right )-960 x^3 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )\right )\right )}{19200 \sqrt {-1+x} \sqrt {x}} \]
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\[\int x^{\frac {3}{2}} \ln \left (-1+4 x +4 \sqrt {\left (-1+x \right ) x}\right )d x\]
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none
Time = 0.34 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.59 \[ \int x^{3/2} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\frac {3840 \, x^{\frac {7}{2}} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) + 15 \, \sqrt {2} x \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) + 15 \, \sqrt {2} x \arctan \left (\frac {3 \, \sqrt {2} \sqrt {x}}{4 \, \sqrt {x^{2} - x}}\right ) - 4 \, {\left (192 \, x^{2} + 376 \, x + 707\right )} \sqrt {x^{2} - x} \sqrt {x} - 4 \, {\left (192 \, x^{3} - 40 \, x^{2} + 15 \, x\right )} \sqrt {x}}{9600 \, x} \]
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Timed out. \[ \int x^{3/2} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\text {Timed out} \]
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\[ \int x^{3/2} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\int { x^{\frac {3}{2}} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right ) \,d x } \]
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Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.71 \[ \int x^{3/2} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\frac {2}{5} \, x^{\frac {5}{2}} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right ) - \frac {2}{25} \, x^{\frac {5}{2}} + \frac {1}{1280} \, \sqrt {2} {\left (\pi - 2 \, \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1\right )}}{3 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right )\right )} + \frac {1}{19200} \, \sqrt {2} {\left (15 i \, \pi + 2828 i \, \sqrt {2} + 30 \, \arctan \left (\frac {2}{3} i \, \sqrt {2}\right )\right )} - \frac {1}{2400} \, {\left (8 \, {\left (24 \, x + 47\right )} x + 707\right )} \sqrt {x - 1} + \frac {1}{60} \, x^{\frac {3}{2}} + \frac {1}{640} \, \sqrt {2} \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) - \frac {1}{160} \, \sqrt {x} \]
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Timed out. \[ \int x^{3/2} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx=\int x^{3/2}\,\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right ) \,d x \]
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