Optimal. Leaf size=71 \[ -\frac {2 \sqrt {x-1}}{3 (3 x+1)}+\frac {4 \sqrt {x}}{3 (3 x+1)}+\frac {8 \tan ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {3}}-\frac {\sqrt {x}}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.22, antiderivative size = 82, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6742, 51, 63, 203, 47} \begin {gather*} -\frac {2 \sqrt {x-1}}{3 (3 x+1)}+\frac {4 \sqrt {x}}{3 (3 x+1)}+\frac {4 \tan ^{-1}\left (\frac {1}{2} \sqrt {3} \sqrt {x-1}\right )}{3 \sqrt {3}}-\frac {4 \tan ^{-1}\left (\sqrt {3} \sqrt {x}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 203
Rule 6742
Rubi steps
\begin {align*} \int \frac {1}{\left (\sqrt {-1+x}+2 \sqrt {x}\right )^2 \sqrt {-1+x}} \, dx &=\int \left (-\frac {8}{3 \sqrt {-1+x} (1+3 x)^2}-\frac {4 \sqrt {x}}{(1+3 x)^2}+\frac {5}{3 \sqrt {-1+x} (1+3 x)}\right ) \, dx\\ &=\frac {5}{3} \int \frac {1}{\sqrt {-1+x} (1+3 x)} \, dx-\frac {8}{3} \int \frac {1}{\sqrt {-1+x} (1+3 x)^2} \, dx-4 \int \frac {\sqrt {x}}{(1+3 x)^2} \, dx\\ &=-\frac {2 \sqrt {-1+x}}{3 (1+3 x)}+\frac {4 \sqrt {x}}{3 (1+3 x)}-\frac {1}{3} \int \frac {1}{\sqrt {-1+x} (1+3 x)} \, dx-\frac {2}{3} \int \frac {1}{\sqrt {x} (1+3 x)} \, dx+\frac {10}{3} \operatorname {Subst}\left (\int \frac {1}{4+3 x^2} \, dx,x,\sqrt {-1+x}\right )\\ &=-\frac {2 \sqrt {-1+x}}{3 (1+3 x)}+\frac {4 \sqrt {x}}{3 (1+3 x)}+\frac {5 \tan ^{-1}\left (\frac {1}{2} \sqrt {3} \sqrt {-1+x}\right )}{3 \sqrt {3}}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{4+3 x^2} \, dx,x,\sqrt {-1+x}\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{1+3 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 \sqrt {-1+x}}{3 (1+3 x)}+\frac {4 \sqrt {x}}{3 (1+3 x)}+\frac {4 \tan ^{-1}\left (\frac {1}{2} \sqrt {3} \sqrt {-1+x}\right )}{3 \sqrt {3}}-\frac {4 \tan ^{-1}\left (\sqrt {3} \sqrt {x}\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 78, normalized size = 1.10 \begin {gather*} \frac {-6 \sqrt {x-1}+12 \sqrt {x}+4 \sqrt {3} (3 x+1) \tan ^{-1}\left (\frac {1}{2} \sqrt {3} \sqrt {x-1}\right )-4 \sqrt {3} (3 x+1) \tan ^{-1}\left (\sqrt {3} \sqrt {x}\right )}{27 x+9} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 71, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {x-1}}{3 (3 x+1)}+\frac {4 \sqrt {x}}{3 (3 x+1)}+\frac {8 \tan ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {3}}-\frac {\sqrt {x}}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 61, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (2 \, \sqrt {3} {\left (3 \, x + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {3} \sqrt {x - 1}\right ) - 2 \, \sqrt {3} {\left (3 \, x + 1\right )} \arctan \left (\sqrt {3} \sqrt {x}\right ) - 3 \, \sqrt {x - 1} + 6 \, \sqrt {x}\right )}}{9 \, {\left (3 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 132, normalized size = 1.86 \begin {gather*} \frac {2}{9} \, \sqrt {3} {\left (\pi - 2 \, \arctan \left (-\frac {\sqrt {3} {\left ({\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} + 1\right )}}{2 \, {\left (\sqrt {x - 1} - \sqrt {x}\right )}}\right )\right )} + \frac {4}{9} \, \sqrt {3} \arctan \left (\frac {1}{2} \, \sqrt {3} \sqrt {x - 1}\right ) - \frac {8 \, {\left (\sqrt {x - 1} - \sqrt {x} + \frac {1}{\sqrt {x - 1} - \sqrt {x}}\right )}}{3 \, {\left (3 \, {\left (\sqrt {x - 1} - \sqrt {x} + \frac {1}{\sqrt {x - 1} - \sqrt {x}}\right )}^{2} + 4\right )}} - \frac {2 \, \sqrt {x - 1}}{3 \, {\left (3 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 67, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {-1+x}}{4 \left (1+3 x \right )}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {-1+x}\, \sqrt {3}}{2}\right )}{9}+\frac {4 \sqrt {x}}{9 \left (\frac {1}{3}+x \right )}-\frac {4 \sqrt {3}\, \arctan \left (\sqrt {x}\, \sqrt {3}\right )}{9}-\frac {5 \sqrt {-1+x}}{36 \left (\frac {1}{3}+x \right )} \end {gather*}
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 {1}{\sqrt {x - 1} {\left (\sqrt {x - 1} + 2 \, \sqrt {x}\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.96, size = 96, normalized size = 1.35 \begin {gather*} \frac {4\,\sqrt {x}}{3\,\left (3\,x+1\right )}-\frac {2\,\sqrt {x-1}}{3\,\left (3\,x+1\right )}+\frac {\sqrt {3}\,\ln \left (\frac {12\,\sqrt {x-1}-\sqrt {3}\,x\,3{}\mathrm {i}+\sqrt {3}\,7{}\mathrm {i}}{3\,x+1}\right )\,2{}\mathrm {i}}{9}+\frac {\sqrt {3}\,\ln \left (\frac {\sqrt {3}-3\,\sqrt {3}\,x+\sqrt {x}\,6{}\mathrm {i}}{x\,3{}\mathrm {i}+1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.56, size = 12, normalized size = 0.17 \begin {gather*} \tilde {\infty } \left (- \frac {2 x}{x - 1} + \frac {2}{x - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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