Optimal. Leaf size=87 \[ -\frac {2 (197-837 x)}{3887 \sqrt {3 x^2-x+2}}-\frac {4 \sqrt {3 x^2-x+2}}{169 (2 x+1)}+\frac {2 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{169 \sqrt {13}} \]
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Rubi [A] time = 0.09, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1646, 806, 724, 206} \[ -\frac {2 (197-837 x)}{3887 \sqrt {3 x^2-x+2}}-\frac {4 \sqrt {3 x^2-x+2}}{169 (2 x+1)}+\frac {2 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{169 \sqrt {13}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 1646
Rubi steps
\begin {align*} \int \frac {1+3 x+4 x^2}{(1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}} \, dx &=-\frac {2 (197-837 x)}{3887 \sqrt {2-x+3 x^2}}+\frac {2}{23} \int \frac {\frac {184}{169}-\frac {230 x}{169}}{(1+2 x)^2 \sqrt {2-x+3 x^2}} \, dx\\ &=-\frac {2 (197-837 x)}{3887 \sqrt {2-x+3 x^2}}-\frac {4 \sqrt {2-x+3 x^2}}{169 (1+2 x)}-\frac {2}{169} \int \frac {1}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx\\ &=-\frac {2 (197-837 x)}{3887 \sqrt {2-x+3 x^2}}-\frac {4 \sqrt {2-x+3 x^2}}{169 (1+2 x)}+\frac {4}{169} \operatorname {Subst}\left (\int \frac {1}{52-x^2} \, dx,x,\frac {9-8 x}{\sqrt {2-x+3 x^2}}\right )\\ &=-\frac {2 (197-837 x)}{3887 \sqrt {2-x+3 x^2}}-\frac {4 \sqrt {2-x+3 x^2}}{169 (1+2 x)}+\frac {2 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{169 \sqrt {13}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 74, normalized size = 0.85 \[ \frac {2 \left (1536 x^2+489 x-289\right )}{3887 (2 x+1) \sqrt {3 x^2-x+2}}+\frac {2 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{169 \sqrt {13}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 106, normalized size = 1.22 \[ \frac {23 \, \sqrt {13} {\left (6 \, x^{3} + x^{2} + 3 \, x + 2\right )} \log \left (\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} - 220 \, x^{2} + 196 \, x - 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 26 \, {\left (1536 \, x^{2} + 489 \, x - 289\right )} \sqrt {3 \, x^{2} - x + 2}}{50531 \, {\left (6 \, x^{3} + x^{2} + 3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 168, normalized size = 1.93 \[ -\frac {2}{50531} \, \sqrt {13} {\left (256 \, \sqrt {13} \sqrt {3} + 23 \, \log \left (\sqrt {13} \sqrt {3} - 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - \frac {2 \, {\left (\frac {\frac {1047}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} + \frac {299}{{\left (2 \, x + 1\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}}{2 \, x + 1} - \frac {768}{\mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}\right )}}{3887 \, \sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3}} + \frac {2 \, \sqrt {13} \log \left (\sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )} - 4\right )}{2197 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 109, normalized size = 1.25 \[ \frac {2 \sqrt {13}\, \arctanh \left (\frac {2 \left (-4 x +\frac {9}{2}\right ) \sqrt {13}}{13 \sqrt {-16 x +12 \left (x +\frac {1}{2}\right )^{2}+5}}\right )}{2197}+\frac {\frac {12 x}{23}-\frac {2}{23}}{\sqrt {3 x^{2}-x +2}}-\frac {1}{169 \sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}}-\frac {82 \left (6 x -1\right )}{3887 \sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}}-\frac {1}{26 \left (x +\frac {1}{2}\right ) \sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 96, normalized size = 1.10 \[ -\frac {2}{2197} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {1536 \, x}{3887 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {279}{3887 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {1}{13 \, {\left (2 \, \sqrt {3 \, x^{2} - x + 2} x + \sqrt {3 \, x^{2} - x + 2}\right )}} \]
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 \[ \int \frac {4\,x^2+3\,x+1}{{\left (2\,x+1\right )}^2\,{\left (3\,x^2-x+2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{2} \left (3 x^{2} - x + 2\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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