Optimal. Leaf size=67 \[ x-2 \sqrt {3} \sqrt {2 x-3}+3 \log \left (x+\sqrt {3} \sqrt {2 x-3}+4\right )+4 \sqrt {6} \tan ^{-1}\left (\frac {\sqrt {6 x-9}+3}{2 \sqrt {6}}\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1628, 634, 618, 204, 628} \[ x-2 \sqrt {3} \sqrt {2 x-3}+3 \log \left (x+\sqrt {3} \sqrt {2 x-3}+4\right )+4 \sqrt {6} \tan ^{-1}\left (\frac {\sqrt {6 x-9}+3}{2 \sqrt {6}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1628
Rubi steps
\begin {align*} \int \frac {1+x}{4+x+\sqrt {-9+6 x}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x \left (15+x^2\right )}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-6+x+\frac {18 (11+x)}{33+6 x+x^2}\right ) \, dx,x,\sqrt {-9+6 x}\right )\\ &=x-2 \sqrt {3} \sqrt {-3+2 x}+6 \operatorname {Subst}\left (\int \frac {11+x}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right )\\ &=x-2 \sqrt {3} \sqrt {-3+2 x}+3 \operatorname {Subst}\left (\int \frac {6+2 x}{33+6 x+x^2} \, dx,x,\sqrt {-9+6 x}\right )+48 \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}+3 \log \left (4+x+\sqrt {3} \sqrt {-3+2 x}\right )-96 \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}+4 \sqrt {6} \tan ^{-1}\left (\frac {3+\sqrt {3} \sqrt {-3+2 x}}{2 \sqrt {6}}\right )+3 \log \left (4+x+\sqrt {3} \sqrt {-3+2 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 56, normalized size = 0.84 \[ x-2 \sqrt {6 x-9}+3 \log \left (x+\sqrt {6 x-9}+4\right )+4 \sqrt {6} \tan ^{-1}\left (\frac {\sqrt {6 x-9}+3}{2 \sqrt {6}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 48, normalized size = 0.72 \[ 4 \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} \sqrt {6 \, x - 9} + \frac {1}{4} \, \sqrt {6}\right ) + x - 2 \, \sqrt {6 \, x - 9} + 3 \, \log \left (x + \sqrt {6 \, x - 9} + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 49, normalized size = 0.73 \[ 4 \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} {\left (\sqrt {6 \, x - 9} + 3\right )}\right ) + x - 2 \, \sqrt {6 \, x - 9} + 3 \, \log \left (6 \, x + 6 \, \sqrt {6 \, x - 9} + 24\right ) - \frac {3}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.78 \[ x +4 \sqrt {6}\, \arctan \left (\frac {\left (2 \sqrt {6 x -9}+6\right ) \sqrt {6}}{24}\right )+3 \ln \left (6 x +24+6 \sqrt {6 x -9}\right )-2 \sqrt {6 x -9}-\frac {3}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.24, size = 49, normalized size = 0.73 \[ 4 \, \sqrt {6} \arctan \left (\frac {1}{12} \, \sqrt {6} {\left (\sqrt {6 \, x - 9} + 3\right )}\right ) + x - 2 \, \sqrt {6 \, x - 9} + 3 \, \log \left (6 \, x + 6 \, \sqrt {6 \, x - 9} + 24\right ) - \frac {3}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.09, size = 102, normalized size = 1.52 \[ x+3\,\ln \left (\left (6\,\sqrt {6\,x-9}+\left (-3+\sqrt {6}\,2{}\mathrm {i}\right )\,\left (2\,\sqrt {6\,x-9}+6\right )+66\right )\,\left (6\,\sqrt {6\,x-9}-\left (3+\sqrt {6}\,2{}\mathrm {i}\right )\,\left (2\,\sqrt {6\,x-9}+6\right )+66\right )\right )+4\,\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,\sqrt {6\,x-9}}{12}+\frac {\sqrt {6}}{4}\right )-2\,\sqrt {6\,x-9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 38.42, size = 58, normalized size = 0.87 \[ x - 2 \sqrt {6 x - 9} + 3 \log {\left (6 x + 6 \sqrt {6 x - 9} + 24 \right )} + 4 \sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} \left (\sqrt {6 x - 9} + 3\right )}{12} \right )} - \frac {3}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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