Optimal. Leaf size=158 \[ -\frac {2 x^{3/2}}{9}-\frac {11 \sqrt {x^2-x}}{12 \sqrt {x}}+\frac {\sqrt {x^2-x} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {x-1}\right )}{24 \sqrt {2} \sqrt {x-1} \sqrt {x}}-\frac {2 \left (x^2-x\right )^{3/2}}{9 x^{3/2}}+\frac {2}{3} x^{3/2} \log \left (4 \sqrt {x^2-x}+4 x-1\right )+\frac {\sqrt {x}}{12}-\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{24 \sqrt {2}} \]
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Rubi [A] time = 0.43, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2537, 2535, 6733, 6742, 203, 1588, 2000, 1146, 444, 50, 63, 204} \[ -\frac {2 x^{3/2}}{9}-\frac {11 \sqrt {x^2-x}}{12 \sqrt {x}}-\frac {2 \left (x^2-x\right )^{3/2}}{9 x^{3/2}}+\frac {2}{3} x^{3/2} \log \left (4 \sqrt {x^2-x}+4 x-1\right )+\frac {\sqrt {x^2-x} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {x-1}\right )}{24 \sqrt {2} \sqrt {x-1} \sqrt {x}}+\frac {\sqrt {x}}{12}-\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{24 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 204
Rule 444
Rule 1146
Rule 1588
Rule 2000
Rule 2535
Rule 2537
Rule 6733
Rule 6742
Rubi steps
\begin {align*} \int \sqrt {x} \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx &=\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} \operatorname {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} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{1+8 x^2} \, dx,x,\sqrt {x}\right )+\frac {2}{9} \operatorname {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{-1-8 x^2} \, dx,x,\sqrt {x}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \sqrt {-x^2+x^4} \, dx,x,\sqrt {x}\right )-\frac {8}{9} \operatorname {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 ) \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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*}
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Mathematica [C] time = 0.65, size = 209, normalized size = 1.32 \[ \frac {1}{576} \left (-128 x^{3/2}+384 x^{3/2} \log \left (4 x+4 \sqrt {(x-1) x}-1\right )-128 \sqrt {(x-1) x} \sqrt {x}+48 \sqrt {x}-\frac {400 \sqrt {(x-1) x}}{\sqrt {x}}+6 i \sqrt {2} \log \left (4 (8 x+1)^2\right )-3 i \sqrt {2} \log \left ((8 x+1) \left (-10 x-6 \sqrt {(x-1) x}+1\right )\right )-3 i \sqrt {2} \log \left ((8 x+1) \left (-10 x+6 \sqrt {(x-1) x}+1\right )\right )-12 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )+12 \sqrt {2} \tan ^{-1}\left (\frac {2 \sqrt {2} \sqrt {(x-1) x}}{3 \sqrt {x}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 100, normalized size = 0.63 \[ \frac {96 \, x^{\frac {5}{2}} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) - 3 \, \sqrt {2} x \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) - 3 \, \sqrt {2} x \arctan \left (\frac {3 \, \sqrt {2} \sqrt {x}}{4 \, \sqrt {x^{2} - x}}\right ) - 4 \, \sqrt {x^{2} - x} {\left (8 \, x + 25\right )} \sqrt {x} - 4 \, {\left (8 \, x^{2} - 3 \, x\right )} \sqrt {x}}{144 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 122, normalized size = 0.77 \[ -\frac {1}{96} \, \sqrt {2} \pi i + \frac {2}{3} \, x^{\frac {3}{2}} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right ) - \frac {1}{96} \, \sqrt {2} {\left (\pi - 2 \, \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1\right )}}{3 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right )\right )} - \frac {1}{36} \, {\left (8 \, x + 25\right )} \sqrt {x - 1} - \frac {2}{9} \, x^{\frac {3}{2}} - \frac {1}{48} \, \sqrt {2} \arctan \left (\frac {2}{3} \, \sqrt {2} i\right ) - \frac {1}{48} \, \sqrt {2} \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) + \frac {25}{36} \, i + \frac {1}{12} \, \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \sqrt {x}\, \ln \left (4 x -1+4 \sqrt {\left (x -1\right ) x}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2}{3} \, x^{\frac {3}{2}} \log \left (4 \, \sqrt {x - 1} \sqrt {x} + 4 \, x - 1\right ) - \frac {4}{9} \, x^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {x} + \int \frac {2 \, x^{2} + x}{3 \, {\left (4 \, x^{\frac {5}{2}} + 4 \, {\left (x^{2} - x\right )} \sqrt {x - 1} - 5 \, x^{\frac {3}{2}} + \sqrt {x}\right )}}\,{d x} + \frac {1}{3} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{3} \, \log \left (\sqrt {x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {x}\,\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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