Optimal. Leaf size=121 \[ \sqrt {x \left (x-\sqrt {x^2-x}\right )} \left (\frac {4}{3 x}-\frac {2 \sqrt {2} \sqrt {\sqrt {x^2-x}+x} \tanh ^{-1}\left (\sqrt {2} \sqrt {\sqrt {x^2-x}+x}\right )}{x}\right )-\frac {4 \sqrt {x^2-x} \sqrt {x \left (x-\sqrt {x^2-x}\right )}}{3 x^2} \]
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Rubi [F] time = 2.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {-x+x^2} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^3} \, dx &=\frac {\sqrt {-x+x^2} \int \frac {\sqrt {-1+x} \sqrt {x^2-x \sqrt {-x+x^2}}}{x^{5/2}} \, dx}{\sqrt {-1+x} \sqrt {x}}\\ &=\frac {\left (2 \sqrt {-x+x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2} \sqrt {x^4-x^2 \sqrt {-x^2+x^4}}}{x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {-1+x} \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 175, normalized size = 1.45 \begin {gather*} \frac {2 \sqrt {(x-1) x} \left (-16 x^2+16 \left (\sqrt {(x-1) x}+1\right ) x-8 \sqrt {(x-1) x}+3 \sqrt {4 x-4 \sqrt {(x-1) x}-2} \left (x-\sqrt {(x-1) x}\right )^{3/2} \log \left (\sqrt {4 x-4 \sqrt {(x-1) x}-2}+2 \sqrt {x-\sqrt {(x-1) x}}\right )-2\right )}{3 \sqrt {x \left (x-\sqrt {(x-1) x}\right )} \left (\sqrt {(x-1) x}-x\right ) \left (-x+\sqrt {(x-1) x}+1\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.22, size = 121, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt {-x+x^2} \sqrt {x \left (x-\sqrt {-x+x^2}\right )}}{3 x^2}+\sqrt {x \left (x-\sqrt {-x+x^2}\right )} \left (\frac {4}{3 x}-\frac {2 \sqrt {2} \sqrt {x+\sqrt {-x+x^2}} \tanh ^{-1}\left (\sqrt {2} \sqrt {x+\sqrt {-x+x^2}}\right )}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 114, normalized size = 0.94 \begin {gather*} \frac {3 \, \sqrt {2} x^{2} \log \left (-\frac {4 \, x^{2} - 2 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (\sqrt {2} x - \sqrt {2} \sqrt {x^{2} - x}\right )} - 4 \, \sqrt {x^{2} - x} x - x}{x}\right ) + 4 \, \sqrt {x^{2} - \sqrt {x^{2} - x} x} {\left (x - \sqrt {x^{2} - x}\right )}}{3 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} - \sqrt {x^{2} - x} x} \sqrt {x^{2} - x}}{x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}-x}\, \sqrt {x^{2}-x \sqrt {x^{2}-x}}}{x^{3}}\, dx\]
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*} \int \frac {\sqrt {x^{2} - \sqrt {x^{2} - x} x} \sqrt {x^{2} - x}}{x^{3}}\,{d x} \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 {\sqrt {x^2-x}\,\sqrt {x^2-x\,\sqrt {x^2-x}}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x - 1\right )} \sqrt {x \left (x - \sqrt {x^{2} - x}\right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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