Optimal. Leaf size=187 \[ -\frac {2 x^{5/2}}{25}+\frac {x^{3/2}}{60}-\frac {2 \left (x^2-x\right )^{3/2}}{25 \sqrt {x}}-\frac {17 \sqrt {x^2-x}}{32 \sqrt {x}}-\frac {\sqrt {x^2-x} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {x-1}\right )}{320 \sqrt {2} \sqrt {x-1} \sqrt {x}}-\frac {71 \left (x^2-x\right )^{3/2}}{300 x^{3/2}}+\frac {2}{5} x^{5/2} \log \left (4 \sqrt {x^2-x}+4 x-1\right )-\frac {\sqrt {x}}{160}+\frac {\tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )}{320 \sqrt {2}} \]
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Rubi [A] time = 0.54, antiderivative size = 187, normalized size of antiderivative = 1.00, 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 50
Rule 63
Rule 203
Rule 444
Rule 1146
Rule 1588
Rule 2000
Rule 2016
Rule 2535
Rule 2537
Rule 6733
Rule 6742
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*}
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Mathematica [C] time = 0.87, 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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fricas [A] time = 0.96, size = 110, normalized size = 0.59 \[ \frac {3840 \, x^{\frac {7}{2}} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) + 15 \, \sqrt {2} x \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) + 15 \, \sqrt {2} x \arctan \left (\frac {3 \, \sqrt {2} \sqrt {x}}{4 \, \sqrt {x^{2} - x}}\right ) - 4 \, {\left (192 \, x^{2} + 376 \, x + 707\right )} \sqrt {x^{2} - x} \sqrt {x} - 4 \, {\left (192 \, x^{3} - 40 \, x^{2} + 15 \, x\right )} \sqrt {x}}{9600 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 132, normalized size = 0.71 \[ \frac {2}{5} \, x^{\frac {5}{2}} \log \left (4 \, x + 4 \, \sqrt {{\left (x - 1\right )} x} - 1\right ) + \frac {1}{1280} \, \sqrt {2} \pi i - \frac {2}{25} \, x^{\frac {5}{2}} + \frac {1}{1280} \, \sqrt {2} {\left (\pi - 2 \, \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1\right )}}{3 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right )\right )} - \frac {1}{2400} \, {\left (8 \, {\left (24 \, x + 47\right )} x + 707\right )} \sqrt {x - 1} + \frac {1}{60} \, x^{\frac {3}{2}} + \frac {1}{640} \, \sqrt {2} \arctan \left (\frac {2}{3} \, \sqrt {2} i\right ) + \frac {1}{640} \, \sqrt {2} \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) + \frac {707}{2400} \, i - \frac {1}{160} \, \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int x^{\frac {3}{2}} \ln \left (4 x -1+4 \sqrt {\left (x -1\right ) x}\right )\, dx \]
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 \[ \frac {2}{5} \, x^{\frac {5}{2}} \log \left (4 \, \sqrt {x - 1} \sqrt {x} + 4 \, x - 1\right ) - \frac {2}{25} \, {\left (2 \, x^{2} + 5\right )} \sqrt {x} - \frac {2}{15} \, x^{\frac {3}{2}} + \int \frac {2 \, x^{\frac {5}{2}} + x^{\frac {3}{2}}}{5 \, {\left (4 \, x^{2} + 4 \, {\left (x^{\frac {3}{2}} - \sqrt {x}\right )} \sqrt {x - 1} - 5 \, x + 1\right )}}\,{d x} + \frac {1}{5} \, \log \left (\sqrt {x} + 1\right ) - \frac {1}{5} \, \log \left (\sqrt {x} - 1\right ) \]
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 \[ \int x^{3/2}\,\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right ) \,d x \]
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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