Optimal. Leaf size=149 \[ -\frac{x^3}{18}+\frac{x^2}{96}-\frac{1}{18} \left (x^2-x\right )^{3/2}+\frac{5}{64} (1-2 x) \sqrt{x^2-x}-\frac{85 \sqrt{x^2-x}}{384}+\frac{1}{3} x^3 \log \left (4 \sqrt{x^2-x}+4 x-1\right )-\frac{\tanh ^{-1}\left (\frac{1-10 x}{6 \sqrt{x^2-x}}\right )}{3072}-\frac{223 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-x}}\right )}{1536}-\frac{x}{384}+\frac{\log (8 x+1)}{3072} \]
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Rubi [A] time = 0.291557, antiderivative size = 149, normalized size of antiderivative = 1., 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 2537
Rule 2535
Rule 6742
Rule 640
Rule 620
Rule 206
Rule 612
Rule 734
Rule 843
Rule 724
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*}
Mathematica [A] time = 0.334861, size = 107, normalized size = 0.72 \[ \frac{-512 x^3+96 x^2-8 \sqrt{(x-1) x} \left (64 x^2+116 x+165\right )+3072 x^3 \log \left (4 x+4 \sqrt{(x-1) x}-1\right )-24 x+6 \log (8 x+1)-669 \log \left (-2 x-2 \sqrt{(x-1) x}+1\right )-3 \log \left (-10 x+6 \sqrt{(x-1) x}+1\right )}{9216} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.007, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\ln \left ( -1+4\,x+4\,\sqrt{ \left ( -1+x \right ) x} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \log \left (4 \, x + 4 \, \sqrt{{\left (x - 1\right )} x} - 1\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.61613, size = 375, normalized size = 2.52 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36025, size = 167, normalized size = 1.12 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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