Optimal. Leaf size=78 \[ \frac {2}{3} \sqrt {3 x^2-x+2}-\frac {\tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{2 \sqrt {13}}-\frac {5 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{6 \sqrt {3}} \]
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Rubi [A] time = 0.10, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1653, 843, 619, 215, 724, 206} \[ \frac {2}{3} \sqrt {3 x^2-x+2}-\frac {\tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{2 \sqrt {13}}-\frac {5 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{6 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 215
Rule 619
Rule 724
Rule 843
Rule 1653
Rubi steps
\begin {align*} \int \frac {1+3 x+4 x^2}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx &=\frac {2}{3} \sqrt {2-x+3 x^2}+\frac {1}{12} \int \frac {16+20 x}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx\\ &=\frac {2}{3} \sqrt {2-x+3 x^2}+\frac {1}{2} \int \frac {1}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx+\frac {5}{6} \int \frac {1}{\sqrt {2-x+3 x^2}} \, dx\\ &=\frac {2}{3} \sqrt {2-x+3 x^2}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+6 x\right )}{6 \sqrt {69}}-\operatorname {Subst}\left (\int \frac {1}{52-x^2} \, dx,x,\frac {9-8 x}{\sqrt {2-x+3 x^2}}\right )\\ &=\frac {2}{3} \sqrt {2-x+3 x^2}-\frac {5 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{6 \sqrt {3}}-\frac {\tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{2 \sqrt {13}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 78, normalized size = 1.00 \[ \frac {2}{3} \sqrt {3 x^2-x+2}-\frac {\tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{2 \sqrt {13}}+\frac {5 \sinh ^{-1}\left (\frac {6 x-1}{\sqrt {23}}\right )}{6 \sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 105, normalized size = 1.35 \[ \frac {5}{36} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + \frac {1}{52} \, \sqrt {13} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + \frac {2}{3} \, \sqrt {3 \, x^{2} - x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 116, normalized size = 1.49 \[ -\frac {5}{18} \, \sqrt {3} \log \left (-6 \, \sqrt {3} x + \sqrt {3} + 6 \, \sqrt {3 \, x^{2} - x + 2}\right ) + \frac {1}{26} \, \sqrt {13} \log \left (-\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {13} - 2 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} - x + 2} \right |}}{2 \, {\left (2 \, \sqrt {3} x - \sqrt {13} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} - x + 2}\right )}}\right ) + \frac {2}{3} \, \sqrt {3 \, x^{2} - x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 0.77 \[ \frac {5 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{18}-\frac {\sqrt {13}\, \arctanh \left (\frac {2 \left (-4 x +\frac {9}{2}\right ) \sqrt {13}}{13 \sqrt {-16 x +12 \left (x +\frac {1}{2}\right )^{2}+5}}\right )}{26}+\frac {2 \sqrt {3 x^{2}-x +2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 67, normalized size = 0.86 \[ \frac {5}{18} \, \sqrt {3} \operatorname {arsinh}\left (\frac {6}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {1}{26} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {2}{3} \, \sqrt {3 \, x^{2} - x + 2} \]
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 \[ \int \frac {4\,x^2+3\,x+1}{\left (2\,x+1\right )\,\sqrt {3\,x^2-x+2}} \,d x \]
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 \[ \int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right ) \sqrt {3 x^{2} - x + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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