Integrand size = 78, antiderivative size = 30 \[ \int \frac {25 x-25 x^2-44 x^3-12 x^4+\left (75+35 x+23 x^2+14 x^3+6 x^4\right ) \log \left (3-x+x^2\right )}{375 x^2+175 x^3+85 x^4+80 x^5+20 x^6} \, dx=\frac {\log \left (3-x+x^2\right )}{5 \left (-x+\frac {x^2}{5+3 x}\right )} \]
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Time = 0.65 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23, number of steps used = 45, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6873, 12, 6860, 723, 814, 648, 632, 210, 642, 907, 1642, 2608, 2605, 2099} \[ \int \frac {25 x-25 x^2-44 x^3-12 x^4+\left (75+35 x+23 x^2+14 x^3+6 x^4\right ) \log \left (3-x+x^2\right )}{375 x^2+175 x^3+85 x^4+80 x^5+20 x^6} \, dx=-\frac {\log \left (x^2-x+3\right )}{5 x}-\frac {\log \left (x^2-x+3\right )}{5 (2 x+5)} \]
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Rule 12
Rule 210
Rule 632
Rule 642
Rule 648
Rule 723
Rule 814
Rule 907
Rule 1642
Rule 2099
Rule 2605
Rule 2608
Rule 6860
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \int \frac {25 x-25 x^2-44 x^3-12 x^4+\left (75+35 x+23 x^2+14 x^3+6 x^4\right ) \log \left (3-x+x^2\right )}{5 x^2 (5+2 x)^2 \left (3-x+x^2\right )} \, dx \\ & = \frac {1}{5} \int \frac {25 x-25 x^2-44 x^3-12 x^4+\left (75+35 x+23 x^2+14 x^3+6 x^4\right ) \log \left (3-x+x^2\right )}{x^2 (5+2 x)^2 \left (3-x+x^2\right )} \, dx \\ & = \frac {1}{5} \int \left (-\frac {25}{(5+2 x)^2 \left (3-x+x^2\right )}+\frac {25}{x (5+2 x)^2 \left (3-x+x^2\right )}-\frac {44 x}{(5+2 x)^2 \left (3-x+x^2\right )}-\frac {12 x^2}{(5+2 x)^2 \left (3-x+x^2\right )}+\frac {\left (25+20 x+6 x^2\right ) \log \left (3-x+x^2\right )}{x^2 (5+2 x)^2}\right ) \, dx \\ & = \frac {1}{5} \int \frac {\left (25+20 x+6 x^2\right ) \log \left (3-x+x^2\right )}{x^2 (5+2 x)^2} \, dx-\frac {12}{5} \int \frac {x^2}{(5+2 x)^2 \left (3-x+x^2\right )} \, dx-5 \int \frac {1}{(5+2 x)^2 \left (3-x+x^2\right )} \, dx+5 \int \frac {1}{x (5+2 x)^2 \left (3-x+x^2\right )} \, dx-\frac {44}{5} \int \frac {x}{(5+2 x)^2 \left (3-x+x^2\right )} \, dx \\ & = \frac {10}{47 (5+2 x)}-\frac {5}{47} \int \frac {7-2 x}{(5+2 x) \left (3-x+x^2\right )} \, dx+\frac {1}{5} \int \left (\frac {\log \left (3-x+x^2\right )}{x^2}+\frac {2 \log \left (3-x+x^2\right )}{(5+2 x)^2}\right ) \, dx-\frac {12}{5} \int \left (\frac {25}{47 (5+2 x)^2}-\frac {170}{2209 (5+2 x)}+\frac {-39+85 x}{2209 \left (3-x+x^2\right )}\right ) \, dx+5 \int \left (\frac {1}{75 x}-\frac {8}{235 (5+2 x)^2}-\frac {856}{55225 (5+2 x)}+\frac {-35-37 x}{6627 \left (3-x+x^2\right )}\right ) \, dx-\frac {44}{5} \int \left (-\frac {10}{47 (5+2 x)^2}-\frac {26}{2209 (5+2 x)}+\frac {72+13 x}{2209 \left (3-x+x^2\right )}\right ) \, dx \\ & = \frac {\log (x)}{15}+\frac {1164 \log (5+2 x)}{11045}+\frac {5 \int \frac {-35-37 x}{3-x+x^2} \, dx}{6627}-\frac {12 \int \frac {-39+85 x}{3-x+x^2} \, dx}{11045}-\frac {44 \int \frac {72+13 x}{3-x+x^2} \, dx}{11045}-\frac {5}{47} \int \left (\frac {48}{47 (5+2 x)}+\frac {37-24 x}{47 \left (3-x+x^2\right )}\right ) \, dx+\frac {1}{5} \int \frac {\log \left (3-x+x^2\right )}{x^2} \, dx+\frac {2}{5} \int \frac {\log \left (3-x+x^2\right )}{(5+2 x)^2} \, dx \\ & = \frac {\log (x)}{15}+\frac {12}{235} \log (5+2 x)-\frac {\log \left (3-x+x^2\right )}{5 x}-\frac {\log \left (3-x+x^2\right )}{5 (5+2 x)}-\frac {5 \int \frac {37-24 x}{3-x+x^2} \, dx}{2209}-\frac {42 \int \frac {1}{3-x+x^2} \, dx}{11045}-\frac {185 \int \frac {-1+2 x}{3-x+x^2} \, dx}{13254}-\frac {286 \int \frac {-1+2 x}{3-x+x^2} \, dx}{11045}-\frac {535 \int \frac {1}{3-x+x^2} \, dx}{13254}-\frac {102 \int \frac {-1+2 x}{3-x+x^2} \, dx}{2209}+\frac {1}{5} \int \frac {-1+2 x}{x \left (3-x+x^2\right )} \, dx+\frac {1}{5} \int \frac {-1+2 x}{15+x+3 x^2+2 x^3} \, dx-\frac {3454 \int \frac {1}{3-x+x^2} \, dx}{11045} \\ & = \frac {\log (x)}{15}+\frac {12}{235} \log (5+2 x)-\frac {5701 \log \left (3-x+x^2\right )}{66270}-\frac {\log \left (3-x+x^2\right )}{5 x}-\frac {\log \left (3-x+x^2\right )}{5 (5+2 x)}+\frac {84 \text {Subst}\left (\int \frac {1}{-11-x^2} \, dx,x,-1+2 x\right )}{11045}+\frac {60 \int \frac {-1+2 x}{3-x+x^2} \, dx}{2209}-\frac {125 \int \frac {1}{3-x+x^2} \, dx}{2209}+\frac {535 \text {Subst}\left (\int \frac {1}{-11-x^2} \, dx,x,-1+2 x\right )}{6627}+\frac {1}{5} \int \left (-\frac {1}{3 x}+\frac {5+x}{3 \left (3-x+x^2\right )}\right ) \, dx+\frac {1}{5} \int \left (-\frac {24}{47 (5+2 x)}+\frac {5+12 x}{47 \left (3-x+x^2\right )}\right ) \, dx+\frac {6908 \text {Subst}\left (\int \frac {1}{-11-x^2} \, dx,x,-1+2 x\right )}{11045} \\ & = \frac {2927 \arctan \left (\frac {1-2 x}{\sqrt {11}}\right )}{33135 \sqrt {11}}+\frac {628 \sqrt {11} \arctan \left (\frac {1-2 x}{\sqrt {11}}\right )}{11045}-\frac {83 \log \left (3-x+x^2\right )}{1410}-\frac {\log \left (3-x+x^2\right )}{5 x}-\frac {\log \left (3-x+x^2\right )}{5 (5+2 x)}+\frac {1}{235} \int \frac {5+12 x}{3-x+x^2} \, dx+\frac {1}{15} \int \frac {5+x}{3-x+x^2} \, dx+\frac {250 \text {Subst}\left (\int \frac {1}{-11-x^2} \, dx,x,-1+2 x\right )}{2209} \\ & = \frac {53}{705} \sqrt {11} \arctan \left (\frac {1-2 x}{\sqrt {11}}\right )-\frac {83 \log \left (3-x+x^2\right )}{1410}-\frac {\log \left (3-x+x^2\right )}{5 x}-\frac {\log \left (3-x+x^2\right )}{5 (5+2 x)}+\frac {6}{235} \int \frac {-1+2 x}{3-x+x^2} \, dx+\frac {1}{30} \int \frac {-1+2 x}{3-x+x^2} \, dx+\frac {11}{235} \int \frac {1}{3-x+x^2} \, dx+\frac {11}{30} \int \frac {1}{3-x+x^2} \, dx \\ & = \frac {53}{705} \sqrt {11} \arctan \left (\frac {1-2 x}{\sqrt {11}}\right )-\frac {\log \left (3-x+x^2\right )}{5 x}-\frac {\log \left (3-x+x^2\right )}{5 (5+2 x)}-\frac {22}{235} \text {Subst}\left (\int \frac {1}{-11-x^2} \, dx,x,-1+2 x\right )-\frac {11}{15} \text {Subst}\left (\int \frac {1}{-11-x^2} \, dx,x,-1+2 x\right ) \\ & = -\frac {\log \left (3-x+x^2\right )}{5 x}-\frac {\log \left (3-x+x^2\right )}{5 (5+2 x)} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {25 x-25 x^2-44 x^3-12 x^4+\left (75+35 x+23 x^2+14 x^3+6 x^4\right ) \log \left (3-x+x^2\right )}{375 x^2+175 x^3+85 x^4+80 x^5+20 x^6} \, dx=-\frac {(5+3 x) \log \left (3-x+x^2\right )}{5 x (5+2 x)} \]
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Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90
method | result | size |
risch | \(-\frac {\left (3 x +5\right ) \ln \left (x^{2}-x +3\right )}{5 \left (5+2 x \right ) x}\) | \(27\) |
norman | \(\frac {-\frac {3 \ln \left (x^{2}-x +3\right ) x}{5}-\ln \left (x^{2}-x +3\right )}{\left (5+2 x \right ) x}\) | \(36\) |
parallelrisch | \(\frac {-6 \ln \left (x^{2}-x +3\right ) x -10 \ln \left (x^{2}-x +3\right )}{10 x \left (5+2 x \right )}\) | \(37\) |
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {25 x-25 x^2-44 x^3-12 x^4+\left (75+35 x+23 x^2+14 x^3+6 x^4\right ) \log \left (3-x+x^2\right )}{375 x^2+175 x^3+85 x^4+80 x^5+20 x^6} \, dx=-\frac {{\left (3 \, x + 5\right )} \log \left (x^{2} - x + 3\right )}{5 \, {\left (2 \, x^{2} + 5 \, x\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {25 x-25 x^2-44 x^3-12 x^4+\left (75+35 x+23 x^2+14 x^3+6 x^4\right ) \log \left (3-x+x^2\right )}{375 x^2+175 x^3+85 x^4+80 x^5+20 x^6} \, dx=\frac {\left (- 3 x - 5\right ) \log {\left (x^{2} - x + 3 \right )}}{10 x^{2} + 25 x} \]
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Time = 0.70 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {25 x-25 x^2-44 x^3-12 x^4+\left (75+35 x+23 x^2+14 x^3+6 x^4\right ) \log \left (3-x+x^2\right )}{375 x^2+175 x^3+85 x^4+80 x^5+20 x^6} \, dx=-\frac {{\left (3 \, x + 5\right )} \log \left (x^{2} - x + 3\right )}{5 \, {\left (2 \, x^{2} + 5 \, x\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {25 x-25 x^2-44 x^3-12 x^4+\left (75+35 x+23 x^2+14 x^3+6 x^4\right ) \log \left (3-x+x^2\right )}{375 x^2+175 x^3+85 x^4+80 x^5+20 x^6} \, dx=-\frac {1}{5} \, {\left (\frac {1}{2 \, x + 5} + \frac {1}{x}\right )} \log \left (x^{2} - x + 3\right ) \]
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Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {25 x-25 x^2-44 x^3-12 x^4+\left (75+35 x+23 x^2+14 x^3+6 x^4\right ) \log \left (3-x+x^2\right )}{375 x^2+175 x^3+85 x^4+80 x^5+20 x^6} \, dx=-\frac {\ln \left (x^2-x+3\right )\,\left (3\,x+5\right )}{5\,x\,\left (2\,x+5\right )} \]
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