Integrand size = 18, antiderivative size = 61 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=\frac {5+4 x}{4 \left (3+6 x+2 x^2\right )^2}-\frac {3+2 x}{2 \left (3+6 x+2 x^2\right )}+\frac {\text {arctanh}\left (\frac {3+2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {652, 628, 632, 212} \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=\frac {\text {arctanh}\left (\frac {2 x+3}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2 x+3}{2 \left (2 x^2+6 x+3\right )}+\frac {4 x+5}{4 \left (2 x^2+6 x+3\right )^2} \]
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Rule 212
Rule 628
Rule 632
Rule 652
Rubi steps \begin{align*} \text {integral}& = \frac {5+4 x}{4 \left (3+6 x+2 x^2\right )^2}+3 \int \frac {1}{\left (3+6 x+2 x^2\right )^2} \, dx \\ & = \frac {5+4 x}{4 \left (3+6 x+2 x^2\right )^2}-\frac {3+2 x}{2 \left (3+6 x+2 x^2\right )}-\int \frac {1}{3+6 x+2 x^2} \, dx \\ & = \frac {5+4 x}{4 \left (3+6 x+2 x^2\right )^2}-\frac {3+2 x}{2 \left (3+6 x+2 x^2\right )}+2 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,6+4 x\right ) \\ & = \frac {5+4 x}{4 \left (3+6 x+2 x^2\right )^2}-\frac {3+2 x}{2 \left (3+6 x+2 x^2\right )}+\frac {\text {arctanh}\left (\frac {3+2 x}{\sqrt {3}}\right )}{\sqrt {3}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.15 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=\frac {1}{12} \left (-\frac {3 \left (13+44 x+36 x^2+8 x^3\right )}{\left (3+6 x+2 x^2\right )^2}-2 \sqrt {3} \log \left (-3+\sqrt {3}-2 x\right )+2 \sqrt {3} \log \left (3+\sqrt {3}+2 x\right )\right ) \]
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Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {-24 x -30}{24 \left (2 x^{2}+6 x +3\right )^{2}}-\frac {4 x +6}{4 \left (2 x^{2}+6 x +3\right )}+\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (4 x +6\right ) \sqrt {3}}{6}\right )}{3}\) | \(56\) |
risch | \(\frac {-2 x^{3}-9 x^{2}-11 x -\frac {13}{4}}{\left (2 x^{2}+6 x +3\right )^{2}}+\frac {\ln \left (3+2 x +\sqrt {3}\right ) \sqrt {3}}{6}-\frac {\ln \left (3+2 x -\sqrt {3}\right ) \sqrt {3}}{6}\) | \(61\) |
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Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.59 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=-\frac {24 \, x^{3} - 2 \, \sqrt {3} {\left (4 \, x^{4} + 24 \, x^{3} + 48 \, x^{2} + 36 \, x + 9\right )} \log \left (\frac {2 \, x^{2} + \sqrt {3} {\left (2 \, x + 3\right )} + 6 \, x + 6}{2 \, x^{2} + 6 \, x + 3}\right ) + 108 \, x^{2} + 132 \, x + 39}{12 \, {\left (4 \, x^{4} + 24 \, x^{3} + 48 \, x^{2} + 36 \, x + 9\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.25 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=\frac {- 8 x^{3} - 36 x^{2} - 44 x - 13}{16 x^{4} + 96 x^{3} + 192 x^{2} + 144 x + 36} - \frac {\sqrt {3} \log {\left (x - \frac {\sqrt {3}}{2} + \frac {3}{2} \right )}}{6} + \frac {\sqrt {3} \log {\left (x + \frac {\sqrt {3}}{2} + \frac {3}{2} \right )}}{6} \]
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Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.10 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (\frac {2 \, x - \sqrt {3} + 3}{2 \, x + \sqrt {3} + 3}\right ) - \frac {8 \, x^{3} + 36 \, x^{2} + 44 \, x + 13}{4 \, {\left (4 \, x^{4} + 24 \, x^{3} + 48 \, x^{2} + 36 \, x + 9\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | 4 \, x - 2 \, \sqrt {3} + 6 \right |}}{{\left | 4 \, x + 2 \, \sqrt {3} + 6 \right |}}\right ) - \frac {8 \, x^{3} + 36 \, x^{2} + 44 \, x + 13}{4 \, {\left (2 \, x^{2} + 6 \, x + 3\right )}^{2}} \]
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Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {-3+2 x}{\left (3+6 x+2 x^2\right )^3} \, dx=\frac {\sqrt {3}\,\mathrm {atanh}\left (\sqrt {3}\,\left (\frac {2\,x}{3}+1\right )\right )}{3}-\frac {\frac {x^3}{2}+\frac {9\,x^2}{4}+\frac {11\,x}{4}+\frac {13}{16}}{x^4+6\,x^3+12\,x^2+9\,x+\frac {9}{4}} \]
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