Integrand size = 92, antiderivative size = 34 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=1+\frac {1}{4} \log \left (x+x^2+\frac {\left (\frac {2+\frac {5}{x^2}-x}{x}+x\right )^2}{x^2}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {2099, 1601} \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=\frac {1}{4} \log \left (x^{10}+x^9+x^8-2 x^7+5 x^6-4 x^5+14 x^4-10 x^3+20 x^2+25\right )-2 \log (x) \]
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Rule 1601
Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2}{x}+\frac {x \left (40-30 x+56 x^2-20 x^3+30 x^4-14 x^5+8 x^6+9 x^7+10 x^8\right )}{4 \left (25+20 x^2-10 x^3+14 x^4-4 x^5+5 x^6-2 x^7+x^8+x^9+x^{10}\right )}\right ) \, dx \\ & = -2 \log (x)+\frac {1}{4} \int \frac {x \left (40-30 x+56 x^2-20 x^3+30 x^4-14 x^5+8 x^6+9 x^7+10 x^8\right )}{25+20 x^2-10 x^3+14 x^4-4 x^5+5 x^6-2 x^7+x^8+x^9+x^{10}} \, dx \\ & = -2 \log (x)+\frac {1}{4} \log \left (25+20 x^2-10 x^3+14 x^4-4 x^5+5 x^6-2 x^7+x^8+x^9+x^{10}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=\frac {1}{4} \left (-8 \log (x)+\log \left (25+20 x^2-10 x^3+14 x^4-4 x^5+5 x^6-2 x^7+x^8+x^9+x^{10}\right )\right ) \]
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Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47
method | result | size |
default | \(\frac {\ln \left (x^{10}+x^{9}+x^{8}-2 x^{7}+5 x^{6}-4 x^{5}+14 x^{4}-10 x^{3}+20 x^{2}+25\right )}{4}-2 \ln \left (x \right )\) | \(50\) |
norman | \(\frac {\ln \left (x^{10}+x^{9}+x^{8}-2 x^{7}+5 x^{6}-4 x^{5}+14 x^{4}-10 x^{3}+20 x^{2}+25\right )}{4}-2 \ln \left (x \right )\) | \(50\) |
risch | \(\frac {\ln \left (x^{10}+x^{9}+x^{8}-2 x^{7}+5 x^{6}-4 x^{5}+14 x^{4}-10 x^{3}+20 x^{2}+25\right )}{4}-2 \ln \left (x \right )\) | \(50\) |
parallelrisch | \(\frac {\ln \left (x^{10}+x^{9}+x^{8}-2 x^{7}+5 x^{6}-4 x^{5}+14 x^{4}-10 x^{3}+20 x^{2}+25\right )}{4}-2 \ln \left (x \right )\) | \(50\) |
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Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=\frac {1}{4} \, \log \left (x^{10} + x^{9} + x^{8} - 2 \, x^{7} + 5 \, x^{6} - 4 \, x^{5} + 14 \, x^{4} - 10 \, x^{3} + 20 \, x^{2} + 25\right ) - 2 \, \log \left (x\right ) \]
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Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=- 2 \log {\left (x \right )} + \frac {\log {\left (x^{10} + x^{9} + x^{8} - 2 x^{7} + 5 x^{6} - 4 x^{5} + 14 x^{4} - 10 x^{3} + 20 x^{2} + 25 \right )}}{4} \]
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Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=\frac {1}{4} \, \log \left (x^{10} + x^{9} + x^{8} - 2 \, x^{7} + 5 \, x^{6} - 4 \, x^{5} + 14 \, x^{4} - 10 \, x^{3} + 20 \, x^{2} + 25\right ) - 2 \, \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=\frac {1}{4} \, \log \left (x^{10} + x^{9} + x^{8} - 2 \, x^{7} + 5 \, x^{6} - 4 \, x^{5} + 14 \, x^{4} - 10 \, x^{3} + 20 \, x^{2} + 25\right ) - 2 \, \log \left ({\left | x \right |}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {-200-120 x^2+50 x^3-56 x^4+12 x^5-10 x^6+2 x^7+x^9+2 x^{10}}{100 x+80 x^3-40 x^4+56 x^5-16 x^6+20 x^7-8 x^8+4 x^9+4 x^{10}+4 x^{11}} \, dx=\frac {\ln \left (x^{10}+x^9+x^8-2\,x^7+5\,x^6-4\,x^5+14\,x^4-10\,x^3+20\,x^2+25\right )}{4}-2\,\ln \left (x\right ) \]
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