Optimal. Leaf size=158 \[ -\frac{2 x^{3/2}}{9}-\frac{11 \sqrt{x^2-x}}{12 \sqrt{x}}-\frac{2 \left (x^2-x\right )^{3/2}}{9 x^{3/2}}+\frac{2}{3} x^{3/2} \log \left (4 \sqrt{x^2-x}+4 x-1\right )+\frac{\sqrt{x^2-x} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{x-1}\right )}{24 \sqrt{2} \sqrt{x-1} \sqrt{x}}+\frac{\sqrt{x}}{12}-\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{24 \sqrt{2}} \]
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Rubi [A] time = 0.431156, antiderivative size = 158, normalized size of antiderivative = 1., 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 2537
Rule 2535
Rule 6733
Rule 6742
Rule 203
Rule 1588
Rule 2000
Rule 1146
Rule 444
Rule 50
Rule 63
Rule 204
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*}
Mathematica [C] time = 0.640654, 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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Maple [F] time = 0.007, size = 0, normalized size = 0. \begin{align*} \int \sqrt{x}\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*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.34709, size = 286, normalized size = 1.81 \begin{align*} \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} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x} \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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