Optimal. Leaf size=151 \[ \frac {32 \sqrt {2} \sqrt {x^2-x} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {x-1}\right )}{3 \sqrt {x-1} \sqrt {x}}+\frac {44}{3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x^2-x}}\right )+\frac {4 \sqrt {x^2-x}}{3 x^{3/2}}-\frac {2 \log \left (4 \sqrt {x^2-x}+4 x-1\right )}{3 x^{3/2}}-\frac {16}{3 \sqrt {x}}-\frac {32}{3} \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right ) \]
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Rubi [A] time = 0.48, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {2537, 2535, 6733, 6742, 203, 1588, 2020, 2008, 2021, 1146, 444, 50, 63} \[ \frac {4 \sqrt {x^2-x}}{3 x^{3/2}}-\frac {2 \log \left (4 \sqrt {x^2-x}+4 x-1\right )}{3 x^{3/2}}+\frac {32 \sqrt {2} \sqrt {x^2-x} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {x-1}\right )}{3 \sqrt {x-1} \sqrt {x}}+\frac {44}{3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {x^2-x}}\right )-\frac {16}{3 \sqrt {x}}-\frac {32}{3} \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 444
Rule 1146
Rule 1588
Rule 2008
Rule 2020
Rule 2021
Rule 2535
Rule 2537
Rule 6733
Rule 6742
Rubi steps
\begin {align*} \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{5/2}} \, dx &=\int \frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x^{5/2}} \, dx\\ &=-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}-\frac {16}{3} \int \frac {1}{x^{3/2} \left (-4 (1+2 x) \sqrt {-x+x^2}+8 \left (-x+x^2\right )\right )} \, dx\\ &=-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}-\frac {32}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (-4 \left (1+2 x^2\right ) \sqrt {-x^2+x^4}+8 \left (-x^2+x^4\right )\right )} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}-\frac {32}{3} \operatorname {Subst}\left (\int \left (-\frac {1}{2 x^2}+\frac {4}{1+8 x^2}-\frac {x^2}{12 \sqrt {-x^2+x^4}}+\frac {\sqrt {-x^2+x^4}}{4 x^4}-\frac {5 \sqrt {-x^2+x^4}}{4 x^2}+\frac {32 \sqrt {-x^2+x^4}}{3 \left (1+8 x^2\right )}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {16}{3 \sqrt {x}}-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}+\frac {8}{9} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-x^2+x^4}} \, dx,x,\sqrt {x}\right )-\frac {8}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{x^4} \, dx,x,\sqrt {x}\right )+\frac {40}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{x^2} \, dx,x,\sqrt {x}\right )-\frac {128}{3} \operatorname {Subst}\left (\int \frac {1}{1+8 x^2} \, dx,x,\sqrt {x}\right )-\frac {1024}{9} \operatorname {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{1+8 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {16}{3 \sqrt {x}}+\frac {4 \sqrt {-x+x^2}}{3 x^{3/2}}+\frac {128 \sqrt {-x+x^2}}{9 \sqrt {x}}-\frac {32}{3} \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-x^2+x^4}} \, dx,x,\sqrt {x}\right )-\frac {40}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-x^2+x^4}} \, dx,x,\sqrt {x}\right )-\frac {\left (1024 \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 {16}{3 \sqrt {x}}+\frac {4 \sqrt {-x+x^2}}{3 x^{3/2}}+\frac {128 \sqrt {-x+x^2}}{9 \sqrt {x}}-\frac {32}{3} \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )+\frac {40}{3} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {\left (512 \sqrt {-x+x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{1+8 x} \, dx,x,x\right )}{9 \sqrt {-1+x} \sqrt {x}}\\ &=-\frac {16}{3 \sqrt {x}}+\frac {4 \sqrt {-x+x^2}}{3 x^{3/2}}-\frac {32}{3} \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )+\frac {44}{3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}+\frac {\left (64 \sqrt {-x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} (1+8 x)} \, dx,x,x\right )}{\sqrt {-1+x} \sqrt {x}}\\ &=-\frac {16}{3 \sqrt {x}}+\frac {4 \sqrt {-x+x^2}}{3 x^{3/2}}-\frac {32}{3} \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )+\frac {44}{3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}+\frac {\left (128 \sqrt {-x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{9+8 x^2} \, dx,x,\sqrt {-1+x}\right )}{\sqrt {-1+x} \sqrt {x}}\\ &=-\frac {16}{3 \sqrt {x}}+\frac {4 \sqrt {-x+x^2}}{3 x^{3/2}}+\frac {32 \sqrt {2} \sqrt {-x+x^2} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {-1+x}\right )}{3 \sqrt {-1+x} \sqrt {x}}-\frac {32}{3} \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )+\frac {44}{3} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{3 x^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.73, size = 204, normalized size = 1.35 \[ \frac {2}{3} \left (\frac {2 \sqrt {(x-1) x}}{x^{3/2}}-\frac {\log \left (4 x+4 \sqrt {(x-1) x}-1\right )}{x^{3/2}}-\frac {8}{\sqrt {x}}+8 i \sqrt {2} \log \left (4 (8 x+1)^2\right )-4 i \sqrt {2} \log \left ((8 x+1) \left (-10 x-6 \sqrt {(x-1) x}+1\right )\right )-4 i \sqrt {2} \log \left ((8 x+1) \left (-10 x+6 \sqrt {(x-1) x}+1\right )\right )-16 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-22 \tan ^{-1}\left (\frac {\sqrt {(x-1) x}}{\sqrt {x}}\right )+16 \sqrt {2} \tan ^{-1}\left (\frac {2 \sqrt {2} \sqrt {(x-1) x}}{3 \sqrt {x}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 108, normalized size = 0.72 \[ -\frac {2 \, {\left (16 \, \sqrt {2} x^{2} \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) + 16 \, \sqrt {2} x^{2} \arctan \left (\frac {3 \, \sqrt {2} \sqrt {x}}{4 \, \sqrt {x^{2} - x}}\right ) - 22 \, x^{2} \arctan \left (\frac {\sqrt {x}}{\sqrt {x^{2} - x}}\right ) + 8 \, x^{\frac {3}{2}} + \sqrt {x} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right ) - 2 \, \sqrt {x^{2} - x} \sqrt {x}\right )}}{3 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 181, normalized size = 1.20 \[ \frac {22}{3} \, \pi - \frac {16}{3} \, \sqrt {2} {\left (\pi - 2 \, \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1\right )}}{3 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right )\right )} - \frac {32}{3} \, \sqrt {2} \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) + \frac {8 \, {\left (\sqrt {x - 1} - \sqrt {x} - \frac {1}{\sqrt {x - 1} - \sqrt {x}}\right )}}{3 \, {\left ({\left (\sqrt {x - 1} - \sqrt {x} - \frac {1}{\sqrt {x - 1} - \sqrt {x}}\right )}^{2} + 4\right )}} - \frac {16}{3 \, \sqrt {x}} - \frac {2 \, \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right )}{3 \, x^{\frac {3}{2}}} - \frac {44}{3} \, \arctan \left (\frac {{\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} - 1}{2 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right ) \]
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 \[ \int \frac {\ln \left (4 x -1+4 \sqrt {\left (x -1\right ) x}\right )}{x^{\frac {5}{2}}}\, dx \]
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 \[ \frac {2}{3 \, \sqrt {x}} - \frac {2 \, \log \left (4 \, \sqrt {x - 1} \sqrt {x} + 4 \, x - 1\right )}{3 \, x^{\frac {3}{2}}} - \frac {2}{9 \, x^{\frac {3}{2}}} - \int \frac {2 \, x^{2} + x}{3 \, {\left (4 \, x^{\frac {11}{2}} - 5 \, x^{\frac {9}{2}} + x^{\frac {7}{2}} + 4 \, {\left (x^{5} - x^{4}\right )} \sqrt {x - 1}\right )}}\,{d x} - \frac {1}{3} \, \log \left (\sqrt {x} + 1\right ) + \frac {1}{3} \, \log \left (\sqrt {x} - 1\right ) \]
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 \[ \int \frac {\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right )}{x^{5/2}} \,d x \]
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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