Optimal. Leaf size=114 \[ -\frac {4 \sqrt {2} \sqrt {-x+x^2} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {-1+x}\right )}{\sqrt {-1+x} \sqrt {x}}+4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-8 \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}} \]
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Rubi [A]
time = 0.21, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 13, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {2617, 2615,
6865, 6874, 209, 1602, 2046, 2033, 1160, 455, 52, 65, 210} \begin {gather*} -\frac {4 \sqrt {2} \sqrt {x^2-x} \text {ArcTan}\left (\frac {2}{3} \sqrt {2} \sqrt {x-1}\right )}{\sqrt {x-1} \sqrt {x}}-8 \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt {x^2-x}}\right )+4 \sqrt {2} \text {ArcTan}\left (2 \sqrt {2} \sqrt {x}\right )-\frac {2 \log \left (4 \sqrt {x^2-x}+4 x-1\right )}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 210
Rule 455
Rule 1160
Rule 1602
Rule 2033
Rule 2046
Rule 2615
Rule 2617
Rule 6865
Rule 6874
Rubi steps
\begin {align*} \int \frac {\log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{x^{3/2}} \, dx &=\int \frac {\log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{x^{3/2}} \, dx\\ &=-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}-16 \int \frac {1}{\sqrt {x} \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 )}{\sqrt {x}}-32 \text {Subst}\left (\int \frac {1}{-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 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}-32 \text {Subst}\left (\int \left (-\frac {1}{2 \left (1+8 x^2\right )}-\frac {x^2}{12 \sqrt {-x^2+x^4}}+\frac {\sqrt {-x^2+x^4}}{4 x^2}+\frac {4 \sqrt {-x^2+x^4}}{3 \left (-1-8 x^2\right )}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}+\frac {8}{3} \text {Subst}\left (\int \frac {x^2}{\sqrt {-x^2+x^4}} \, dx,x,\sqrt {x}\right )-8 \text {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{x^2} \, dx,x,\sqrt {x}\right )+16 \text {Subst}\left (\int \frac {1}{1+8 x^2} \, dx,x,\sqrt {x}\right )-\frac {128}{3} \text {Subst}\left (\int \frac {\sqrt {-x^2+x^4}}{-1-8 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {16 \sqrt {-x+x^2}}{3 \sqrt {x}}+4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}+8 \text {Subst}\left (\int \frac {1}{\sqrt {-x^2+x^4}} \, dx,x,\sqrt {x}\right )-\frac {\left (128 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {x \sqrt {-1+x^2}}{-1-8 x^2} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-1+x} \sqrt {x}}\\ &=-\frac {16 \sqrt {-x+x^2}}{3 \sqrt {x}}+4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}-8 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {\left (64 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x}}{-1-8 x} \, dx,x,x\right )}{3 \sqrt {-1+x} \sqrt {x}}\\ &=4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-8 \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}+\frac {\left (24 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{(-1-8 x) \sqrt {-1+x}} \, dx,x,x\right )}{\sqrt {-1+x} \sqrt {x}}\\ &=4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-8 \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}+\frac {\left (48 \sqrt {-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{-9-8 x^2} \, dx,x,\sqrt {-1+x}\right )}{\sqrt {-1+x} \sqrt {x}}\\ &=-\frac {4 \sqrt {2} \sqrt {-x+x^2} \tan ^{-1}\left (\frac {2}{3} \sqrt {2} \sqrt {-1+x}\right )}{\sqrt {-1+x} \sqrt {x}}+4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )-8 \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-x+x^2}}\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {-x+x^2}\right )}{\sqrt {x}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.11, size = 177, normalized size = 1.55 \begin {gather*} 4 \sqrt {2} \tan ^{-1}\left (2 \sqrt {2} \sqrt {x}\right )+8 \tan ^{-1}\left (\frac {\sqrt {(-1+x) x}}{\sqrt {x}}\right )-4 \sqrt {2} \tan ^{-1}\left (\frac {2 \sqrt {2} \sqrt {(-1+x) x}}{3 \sqrt {x}}\right )-2 i \sqrt {2} \log \left (4 (1+8 x)^2\right )+i \sqrt {2} \log \left ((1+8 x) \left (1-10 x-6 \sqrt {(-1+x) x}\right )\right )-\frac {2 \log \left (-1+4 x+4 \sqrt {(-1+x) x}\right )}{\sqrt {x}}+i \sqrt {2} \log \left ((1+8 x) \left (1-10 x+6 \sqrt {(-1+x) x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (-1+4 x +4 \sqrt {\left (-1+x \right ) x}\right )}{x^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 84, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (2 \, \sqrt {2} x \arctan \left (2 \, \sqrt {2} \sqrt {x}\right ) + 2 \, \sqrt {2} x \arctan \left (\frac {3 \, \sqrt {2} \sqrt {x}}{4 \, \sqrt {x^{2} - x}}\right ) - 4 \, x \arctan \left (\frac {\sqrt {x}}{\sqrt {x^{2} - x}}\right ) - \sqrt {x} \log \left (4 \, x + 4 \, \sqrt {x^{2} - x} - 1\right )\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (4\,x+4\,\sqrt {x\,\left (x-1\right )}-1\right )}{x^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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