Integrand size = 110, antiderivative size = 29 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=2+x+\log \left (\frac {6}{x}+\frac {5}{-\frac {2 x}{5}+(1+x)^2}-\log (x)\right ) \]
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\[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=\int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-150-355 x-359 x^2+40 x^3+375 x^4+125 x^5-\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{x \left (5+8 x+5 x^2\right ) \left (30+73 x+30 x^2-5 x \log (x)-8 x^2 \log (x)-5 x^3 \log (x)\right )} \, dx \\ & = \int \left (1+\frac {150+505 x+964 x^2+844 x^3+230 x^4+25 x^5}{x \left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}\right ) \, dx \\ & = x+\int \frac {150+505 x+964 x^2+844 x^3+230 x^4+25 x^5}{x \left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx \\ & = x+\int \left (\frac {103}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)}+\frac {30}{x \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}+\frac {38 x}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)}+\frac {5 x^2}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)}-\frac {50 (5+4 x)}{\left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}\right ) \, dx \\ & = x+5 \int \frac {x^2}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+30 \int \frac {1}{x \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+38 \int \frac {x}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx-50 \int \frac {5+4 x}{\left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+103 \int \frac {1}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx \\ & = x+5 \int \frac {x^2}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+30 \int \frac {1}{x \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+38 \int \frac {x}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx-50 \int \left (\frac {5}{\left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}+\frac {4 x}{\left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}\right ) \, dx+103 \int \frac {1}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx \\ & = x+5 \int \frac {x^2}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+30 \int \frac {1}{x \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+38 \int \frac {x}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+103 \int \frac {1}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx-200 \int \frac {x}{\left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx-250 \int \frac {1}{\left (5+8 x+5 x^2\right ) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx \\ & = x+5 \int \frac {x^2}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+30 \int \frac {1}{x \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+38 \int \frac {x}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+103 \int \frac {1}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx-200 \int \left (\frac {1+\frac {4 i}{3}}{((8-6 i)+10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}+\frac {1-\frac {4 i}{3}}{((8+6 i)+10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}\right ) \, dx-250 \int \left (\frac {5 i}{3 ((-8+6 i)-10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}+\frac {5 i}{3 ((8+6 i)+10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )}\right ) \, dx \\ & = x-\frac {1250}{3} i \int \frac {1}{((-8+6 i)-10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx-\frac {1250}{3} i \int \frac {1}{((8+6 i)+10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+5 \int \frac {x^2}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+30 \int \frac {1}{x \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx+38 \int \frac {x}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx+103 \int \frac {1}{-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)} \, dx-\left (200-\frac {800 i}{3}\right ) \int \frac {1}{((8+6 i)+10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx-\left (200+\frac {800 i}{3}\right ) \int \frac {1}{((8-6 i)+10 x) \left (-30-73 x-30 x^2+5 x \log (x)+8 x^2 \log (x)+5 x^3 \log (x)\right )} \, dx \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=x-\log \left (x \left (5+8 x+5 x^2\right )\right )+\log \left (30+73 x+30 x^2-5 x \log (x)-8 x^2 \log (x)-5 x^3 \log (x)\right ) \]
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Time = 0.38 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17
method | result | size |
risch | \(x +\ln \left (\ln \left (x \right )-\frac {30 x^{2}+73 x +30}{x \left (5 x^{2}+8 x +5\right )}\right )\) | \(34\) |
parallelrisch | \(-\ln \left (x^{2}+\frac {8}{5} x +1\right )+\ln \left (x^{3} \ln \left (x \right )+\frac {8 x^{2} \ln \left (x \right )}{5}-6 x^{2}+x \ln \left (x \right )-\frac {73 x}{5}-6\right )+x -\ln \left (x \right )\) | \(46\) |
default | \(-\ln \left (x \right )+x -\ln \left (5 x^{2}+8 x +5\right )+\ln \left (5 x^{3} \ln \left (x \right )+8 x^{2} \ln \left (x \right )+5 x \ln \left (x \right )-30 x^{2}-73 x -30\right )\) | \(50\) |
norman | \(-\ln \left (x \right )+x -\ln \left (5 x^{2}+8 x +5\right )+\ln \left (5 x^{3} \ln \left (x \right )+8 x^{2} \ln \left (x \right )+5 x \ln \left (x \right )-30 x^{2}-73 x -30\right )\) | \(50\) |
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Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=x + \log \left (-\frac {30 \, x^{2} - {\left (5 \, x^{3} + 8 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 73 \, x + 30}{5 \, x^{3} + 8 \, x^{2} + 5 \, x}\right ) \]
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Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=x + \log {\left (\frac {- 30 x^{2} - 73 x - 30}{5 x^{3} + 8 x^{2} + 5 x} + \log {\left (x \right )} \right )} \]
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Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=x + \log \left (-\frac {30 \, x^{2} - {\left (5 \, x^{3} + 8 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 73 \, x + 30}{5 \, x^{3} + 8 \, x^{2} + 5 \, x}\right ) \]
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Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=x + \log \left (5 \, x^{3} \log \left (x\right ) + 8 \, x^{2} \log \left (x\right ) - 30 \, x^{2} + 5 \, x \log \left (x\right ) - 73 \, x - 30\right ) - \log \left (5 \, x^{2} + 8 \, x + 5\right ) - \log \left (x\right ) \]
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Timed out. \[ \int \frac {150+355 x+359 x^2-40 x^3-375 x^4-125 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)}{-150 x-605 x^2-884 x^3-605 x^4-150 x^5+\left (25 x^2+80 x^3+114 x^4+80 x^5+25 x^6\right ) \log (x)} \, dx=\int -\frac {355\,x+\ln \left (x\right )\,\left (25\,x^6+80\,x^5+114\,x^4+80\,x^3+25\,x^2\right )+359\,x^2-40\,x^3-375\,x^4-125\,x^5+150}{150\,x-\ln \left (x\right )\,\left (25\,x^6+80\,x^5+114\,x^4+80\,x^3+25\,x^2\right )+605\,x^2+884\,x^3+605\,x^4+150\,x^5} \,d x \]
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