Optimal. Leaf size=71 \[ -x+2 \sqrt {3} \sqrt {2 x-3}+10 \log \left (x+\sqrt {3} \sqrt {2 x-3}+4\right )-21 \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {6 x-9}+3}{2 \sqrt {6}}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1628, 634, 618, 204, 628} \[ -x+2 \sqrt {3} \sqrt {2 x-3}+10 \log \left (x+\sqrt {3} \sqrt {2 x-3}+4\right )-21 \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {6 x-9}+3}{2 \sqrt {6}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1628
Rubi steps
\begin {align*} \int \frac {12-x}{4+x+\sqrt {-9+6 x}} \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {x \left (-63+x^2\right )}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right )\right )\\ &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \left (-6+x+\frac {6 (33-10 x)}{33+6 x+x^2}\right ) \, dx,x,\sqrt {-9+6 x}\right )\right )\\ &=-x+2 \sqrt {3} \sqrt {-3+2 x}-2 \operatorname {Subst}\left (\int \frac {33-10 x}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right )\\ &=-x+2 \sqrt {3} \sqrt {-3+2 x}+10 \operatorname {Subst}\left (\int \frac {6+2 x}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right )-126 \operatorname {Subst}\left (\int \frac {1}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right )\\ &=-x+2 \sqrt {3} \sqrt {-3+2 x}+10 \log \left (4+x+\sqrt {3} \sqrt {-3+2 x}\right )+252 \operatorname {Subst}\left (\int \frac {1}{-96-x^2} \, dx,x,6+2 \sqrt {-9+6 x}\right )\\ &=-x+2 \sqrt {3} \sqrt {-3+2 x}-21 \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {3+\sqrt {3} \sqrt {-3+2 x}}{2 \sqrt {6}}\right )+10 \log \left (4+x+\sqrt {3} \sqrt {-3+2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 0.85 \[ -x+2 \sqrt {6 x-9}+10 \log \left (x+\sqrt {6 x-9}+4\right )-21 \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {6 x-9}+3}{2 \sqrt {6}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 59, normalized size = 0.83 \[ -\frac {21}{2} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {1}{12} \, \sqrt {3} \sqrt {2} \sqrt {6 \, x - 9} + \frac {1}{4} \, \sqrt {3} \sqrt {2}\right ) - x + 2 \, \sqrt {6 \, x - 9} + 10 \, \log \left (x + \sqrt {6 \, x - 9} + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 51, normalized size = 0.72 \[ -\frac {21}{2} \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} {\left (\sqrt {6 \, x - 9} + 3\right )}\right ) - x + 2 \, \sqrt {6 \, x - 9} + 10 \, \log \left (6 \, x + 6 \, \sqrt {6 \, x - 9} + 24\right ) + \frac {3}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 54, normalized size = 0.76 \[ -x -\frac {21 \sqrt {6}\, \arctan \left (\frac {\left (2 \sqrt {6 x -9}+6\right ) \sqrt {6}}{24}\right )}{2}+10 \ln \left (6 x +24+6 \sqrt {6 x -9}\right )+2 \sqrt {6 x -9}+\frac {3}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.08, size = 51, normalized size = 0.72 \[ -\frac {21}{2} \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} {\left (\sqrt {6 \, x - 9} + 3\right )}\right ) - x + 2 \, \sqrt {6 \, x - 9} + 10 \, \log \left (6 \, x + 6 \, \sqrt {6 \, x - 9} + 24\right ) + \frac {3}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 118, normalized size = 1.66 \[ 2\,\sqrt {6\,x-9}+10\,\ln \left (\left (\left (2\,\sqrt {6\,x-9}+6\right )\,\left (-10+\frac {\sqrt {2}\,\sqrt {3}\,21{}\mathrm {i}}{4}\right )+20\,\sqrt {6\,x-9}-66\right )\,\left (\left (2\,\sqrt {6\,x-9}+6\right )\,\left (10+\frac {\sqrt {2}\,\sqrt {3}\,21{}\mathrm {i}}{4}\right )-20\,\sqrt {6\,x-9}+66\right )\right )-x-\frac {21\,\sqrt {2}\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {3}}{4}+\frac {\sqrt {2}\,\sqrt {3}\,\sqrt {6\,x-9}}{12}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 73.42, size = 60, normalized size = 0.85 \[ - x + 2 \sqrt {6 x - 9} + 10 \log {\left (6 x + 6 \sqrt {6 x - 9} + 24 \right )} - \frac {21 \sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} \left (\sqrt {6 x - 9} + 3\right )}{12} \right )}}{2} + \frac {3}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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