Optimal. Leaf size=187 \[ -\frac{2 x^{5/2}}{25}+\frac{x^{3/2}}{60}-\frac{2 \left (x^2-x\right )^{3/2}}{25 \sqrt{x}}-\frac{17 \sqrt{x^2-x}}{32 \sqrt{x}}-\frac{71 \left (x^2-x\right )^{3/2}}{300 x^{3/2}}+\frac{2}{5} x^{5/2} \log \left (4 \sqrt{x^2-x}+4 x-1\right )-\frac{\sqrt{x^2-x} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{x-1}\right )}{320 \sqrt{2} \sqrt{x-1} \sqrt{x}}-\frac{\sqrt{x}}{160}+\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{320 \sqrt{2}} \]
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Rubi [A] time = 0.541197, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 15, 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, 2016, 1146, 444, 50, 63} \[ -\frac{2 x^{5/2}}{25}+\frac{x^{3/2}}{60}-\frac{2 \left (x^2-x\right )^{3/2}}{25 \sqrt{x}}-\frac{17 \sqrt{x^2-x}}{32 \sqrt{x}}-\frac{71 \left (x^2-x\right )^{3/2}}{300 x^{3/2}}+\frac{2}{5} x^{5/2} \log \left (4 \sqrt{x^2-x}+4 x-1\right )-\frac{\sqrt{x^2-x} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{x-1}\right )}{320 \sqrt{2} \sqrt{x-1} \sqrt{x}}-\frac{\sqrt{x}}{160}+\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{320 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 2537
Rule 2535
Rule 6733
Rule 6742
Rule 203
Rule 1588
Rule 2000
Rule 2016
Rule 1146
Rule 444
Rule 50
Rule 63
Rubi steps
\begin{align*} \int x^{3/2} \log \left (-1+4 x+4 \sqrt{(-1+x) x}\right ) \, dx &=\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} \operatorname{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} \operatorname{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} \operatorname{Subst}\left (\int \frac{1}{1+8 x^2} \, dx,x,\sqrt{x}\right )+\frac{1}{60} \operatorname{Subst}\left (\int \frac{\sqrt{-x^2+x^4}}{1+8 x^2} \, dx,x,\sqrt{x}\right )-\frac{2}{5} \operatorname{Subst}\left (\int x^2 \sqrt{-x^2+x^4} \, dx,x,\sqrt{x}\right )-\frac{8}{15} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-x^2+x^4}} \, dx,x,\sqrt{x}\right )-\frac{11}{20} \operatorname{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} \operatorname{Subst}\left (\int \sqrt{-x^2+x^4} \, dx,x,\sqrt{x}\right )+\frac{\sqrt{-x+x^2} \operatorname{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} \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [C] time = 0.828841, size = 232, normalized size = 1.24 \[ \frac{-3072 x^{5/2}-3072 \sqrt{(x-1) x} x^{3/2}+640 x^{3/2}+15360 x^{5/2} \log \left (4 x+4 \sqrt{(x-1) x}-1\right )-6016 \sqrt{(x-1) x} \sqrt{x}-240 \sqrt{x}-\frac{11312 \sqrt{(x-1) x}}{\sqrt{x}}-30 i \sqrt{2} \log \left (4 (8 x+1)^2\right )+15 i \sqrt{2} \log \left ((8 x+1) \left (-10 x-6 \sqrt{(x-1) x}+1\right )\right )+15 i \sqrt{2} \log \left ((8 x+1) \left (-10 x+6 \sqrt{(x-1) x}+1\right )\right )+60 \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )-60 \sqrt{2} \tan ^{-1}\left (\frac{2 \sqrt{2} \sqrt{(x-1) x}}{3 \sqrt{x}}\right )}{38400} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.006, size = 0, normalized size = 0. \begin{align*} \int{x}^{{\frac{3}{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*} \frac{2}{5} \, x^{\frac{5}{2}} \log \left (4 \, \sqrt{x - 1} \sqrt{x} + 4 \, x - 1\right ) - \frac{2}{25} \,{\left (2 \, x^{2} + 5\right )} \sqrt{x} - \frac{2}{15} \, x^{\frac{3}{2}} + \int \frac{2 \, x^{\frac{5}{2}} + x^{\frac{3}{2}}}{5 \,{\left (4 \, x^{2} + 4 \,{\left (x^{\frac{3}{2}} - \sqrt{x}\right )} \sqrt{x - 1} - 5 \, x + 1\right )}}\,{d x} + \frac{1}{5} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{5} \, \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 = 2.25007, size = 327, normalized size = 1.75 \begin{align*} \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} \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 x^{\frac{3}{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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