Optimal. Leaf size=149 \[ -\frac {x^3}{18}+\frac {x^2}{96}-\frac {1}{18} \left (x^2-x\right )^{3/2}+\frac {5}{64} (1-2 x) \sqrt {x^2-x}-\frac {85 \sqrt {x^2-x}}{384}-\frac {\tanh ^{-1}\left (\frac {1-10 x}{6 \sqrt {x^2-x}}\right )}{3072}-\frac {223 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-x}}\right )}{1536}+\frac {1}{3} x^3 \log \left (4 \sqrt {x^2-x}+4 x-1\right )-\frac {x}{384}+\frac {\log (8 x+1)}{3072} \]
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Rubi [A] time = 0.29, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2537, 2535, 6742, 640, 620, 206, 612, 734, 843, 724} \[ -\frac {x^3}{18}+\frac {x^2}{96}-\frac {1}{18} \left (x^2-x\right )^{3/2}+\frac {5}{64} (1-2 x) \sqrt {x^2-x}-\frac {85 \sqrt {x^2-x}}{384}+\frac {1}{3} x^3 \log \left (4 \sqrt {x^2-x}+4 x-1\right )-\frac {\tanh ^{-1}\left (\frac {1-10 x}{6 \sqrt {x^2-x}}\right )}{3072}-\frac {223 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-x}}\right )}{1536}-\frac {x}{384}+\frac {\log (8 x+1)}{3072} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 640
Rule 724
Rule 734
Rule 843
Rule 2535
Rule 2537
Rule 6742
Rubi steps
\begin {align*} \int x^2 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right ) \, dx &=\int x^2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right ) \, dx\\ &=\frac {1}{3} x^3 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {8}{3} \int \frac {x^3}{-4 (1+2 x) \sqrt {-x+x^2}+8 \left (-x+x^2\right )} \, dx\\ &=\frac {1}{3} x^3 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {8}{3} \int \left (-\frac {1}{1024}+\frac {x}{128}-\frac {x^2}{16}+\frac {1}{1024 (1+8 x)}-\frac {x}{12 \sqrt {-x+x^2}}-\frac {11}{128} \sqrt {-x+x^2}-\frac {1}{16} x \sqrt {-x+x^2}+\frac {\sqrt {-x+x^2}}{384 (1+8 x)}\right ) \, dx\\ &=-\frac {x}{384}+\frac {x^2}{96}-\frac {x^3}{18}+\frac {\log (1+8 x)}{3072}+\frac {1}{3} x^3 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )+\frac {1}{144} \int \frac {\sqrt {-x+x^2}}{1+8 x} \, dx-\frac {1}{6} \int x \sqrt {-x+x^2} \, dx-\frac {2}{9} \int \frac {x}{\sqrt {-x+x^2}} \, dx-\frac {11}{48} \int \sqrt {-x+x^2} \, dx\\ &=-\frac {x}{384}+\frac {x^2}{96}-\frac {x^3}{18}-\frac {85}{384} \sqrt {-x+x^2}+\frac {11}{192} (1-2 x) \sqrt {-x+x^2}-\frac {1}{18} \left (-x+x^2\right )^{3/2}+\frac {\log (1+8 x)}{3072}+\frac {1}{3} x^3 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )-\frac {\int \frac {-1+10 x}{(1+8 x) \sqrt {-x+x^2}} \, dx}{2304}+\frac {11}{384} \int \frac {1}{\sqrt {-x+x^2}} \, dx-\frac {1}{12} \int \sqrt {-x+x^2} \, dx-\frac {1}{9} \int \frac {1}{\sqrt {-x+x^2}} \, dx\\ &=-\frac {x}{384}+\frac {x^2}{96}-\frac {x^3}{18}-\frac {85}{384} \sqrt {-x+x^2}+\frac {5}{64} (1-2 x) \sqrt {-x+x^2}-\frac {1}{18} \left (-x+x^2\right )^{3/2}+\frac {\log (1+8 x)}{3072}+\frac {1}{3} x^3 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )-\frac {5 \int \frac {1}{\sqrt {-x+x^2}} \, dx}{9216}+\frac {\int \frac {1}{(1+8 x) \sqrt {-x+x^2}} \, dx}{1024}+\frac {1}{96} \int \frac {1}{\sqrt {-x+x^2}} \, dx+\frac {11}{192} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )-\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )\\ &=-\frac {x}{384}+\frac {x^2}{96}-\frac {x^3}{18}-\frac {85}{384} \sqrt {-x+x^2}+\frac {5}{64} (1-2 x) \sqrt {-x+x^2}-\frac {1}{18} \left (-x+x^2\right )^{3/2}-\frac {95}{576} \tanh ^{-1}\left (\frac {x}{\sqrt {-x+x^2}}\right )+\frac {\log (1+8 x)}{3072}+\frac {1}{3} x^3 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )-\frac {5 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )}{4608}-\frac {1}{512} \operatorname {Subst}\left (\int \frac {1}{36-x^2} \, dx,x,\frac {1-10 x}{\sqrt {-x+x^2}}\right )+\frac {1}{48} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-x+x^2}}\right )\\ &=-\frac {x}{384}+\frac {x^2}{96}-\frac {x^3}{18}-\frac {85}{384} \sqrt {-x+x^2}+\frac {5}{64} (1-2 x) \sqrt {-x+x^2}-\frac {1}{18} \left (-x+x^2\right )^{3/2}-\frac {\tanh ^{-1}\left (\frac {1-10 x}{6 \sqrt {-x+x^2}}\right )}{3072}-\frac {223 \tanh ^{-1}\left (\frac {x}{\sqrt {-x+x^2}}\right )}{1536}+\frac {\log (1+8 x)}{3072}+\frac {1}{3} x^3 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.35, size = 107, normalized size = 0.72 \[ \frac {-512 x^3+3072 x^3 \log \left (4 x+4 \sqrt {(x-1) x}-1\right )+96 x^2-8 \sqrt {(x-1) x} \left (64 x^2+116 x+165\right )-24 x+6 \log (8 x+1)-669 \log \left (-2 x-2 \sqrt {(x-1) x}+1\right )-3 \log \left (-10 x+6 \sqrt {(x-1) x}+1\right )}{9216} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 124, normalized size = 0.83 \[ -\frac {1}{18} \, x^{3} + \frac {1}{96} \, x^{2} + \frac {1}{3} \, {\left (x^{3} + 1\right )} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) - \frac {1}{1152} \, {\left (64 \, x^{2} + 116 \, x + 165\right )} \sqrt {x^{2} - x} - \frac {1}{384} \, x - \frac {511}{3072} \, \log \left (8 \, x + 1\right ) + \frac {245}{1024} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} + 1\right ) - \frac {511}{3072} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} - x} - 1\right ) + \frac {511}{3072} \, \log \left (-4 \, x + 4 \, \sqrt {x^{2} - x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 124, normalized size = 0.83 \[ \frac {1}{3} \, x^{3} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right ) - \frac {1}{18} \, x^{3} + \frac {1}{96} \, x^{2} - \frac {1}{1152} \, {\left (4 \, {\left (16 \, x + 29\right )} x + 165\right )} \sqrt {x^{2} - x} - \frac {1}{384} \, x + \frac {1}{3072} \, \log \left ({\left | 8 \, x + 1 \right |}\right ) + \frac {223}{3072} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} + 1 \right |}\right ) + \frac {1}{3072} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x} - 1 \right |}\right ) - \frac {1}{3072} \, \log \left ({\left | -4 \, x + 4 \, \sqrt {x^{2} - x} + 1 \right |}\right ) \]
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 \[ \int x^{2} \ln \left (4 x -1+4 \sqrt {\left (x -1\right ) x}\right )\, dx \]
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 \[ \int x^{2} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right )\,{d x} \]
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 \[ \int x^2\,\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right ) \,d x \]
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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