Integrand size = 56, antiderivative size = 29 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=-4-x+(9-x)^2 x-x \log \left (7+\frac {3}{5-x}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90, number of steps used = 24, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6860, 630, 31, 646, 717, 715, 2535} \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=x^3-18 x^2+80 x-\frac {38}{7} \log (38-7 x)+\frac {1}{7} (38-7 x) \log \left (\frac {38-7 x}{5-x}\right )+\frac {38}{7} \log (5-x) \]
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Rule 31
Rule 630
Rule 646
Rule 715
Rule 717
Rule 2535
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {15200}{190-73 x+7 x^2}-\frac {12683 x}{190-73 x+7 x^2}+\frac {3758 x^2}{190-73 x+7 x^2}-\frac {471 x^3}{190-73 x+7 x^2}+\frac {21 x^4}{190-73 x+7 x^2}-\log \left (\frac {-38+7 x}{-5+x}\right )\right ) \, dx \\ & = 21 \int \frac {x^4}{190-73 x+7 x^2} \, dx-471 \int \frac {x^3}{190-73 x+7 x^2} \, dx+3758 \int \frac {x^2}{190-73 x+7 x^2} \, dx-12683 \int \frac {x}{190-73 x+7 x^2} \, dx+15200 \int \frac {1}{190-73 x+7 x^2} \, dx-\int \log \left (\frac {-38+7 x}{-5+x}\right ) \, dx \\ & = \frac {3758 x}{7}+\frac {1}{7} (38-7 x) \log \left (\frac {38-7 x}{5-x}\right )+\frac {3}{7} \int \frac {1}{-5+x} \, dx+21 \int \left (\frac {3999}{343}+\frac {73 x}{49}+\frac {x^2}{7}-\frac {759810-194837 x}{343 \left (190-73 x+7 x^2\right )}\right ) \, dx-471 \int \left (\frac {73}{49}+\frac {x}{7}-\frac {13870-3999 x}{49 \left (190-73 x+7 x^2\right )}\right ) \, dx+\frac {3758}{7} \int \frac {-190+73 x}{190-73 x+7 x^2} \, dx+\frac {106400}{3} \int \frac {1}{-38+7 x} \, dx-\frac {106400}{3} \int \frac {1}{-35+7 x} \, dx+\frac {443905}{3} \int \frac {1}{-35+7 x} \, dx-\frac {481954}{3} \int \frac {1}{-38+7 x} \, dx \\ & = 80 x-18 x^2+x^3-\frac {375554}{21} \log (38-7 x)+\frac {1}{7} (38-7 x) \log \left (\frac {38-7 x}{5-x}\right )+\frac {337514}{21} \log (5-x)-\frac {3}{49} \int \frac {759810-194837 x}{190-73 x+7 x^2} \, dx+\frac {471}{49} \int \frac {13870-3999 x}{190-73 x+7 x^2} \, dx-\frac {657650}{3} \int \frac {1}{-35+7 x} \, dx+\frac {5426552}{21} \int \frac {1}{-38+7 x} \, dx \\ & = 80 x-18 x^2+x^3+\frac {932558}{49} \log (38-7 x)+\frac {1}{7} (38-7 x) \log \left (\frac {38-7 x}{5-x}\right )-\frac {106712}{7} \log (5-x)-30625 \int \frac {1}{-35+7 x} \, dx+\frac {2085136}{49} \int \frac {1}{-38+7 x} \, dx+137375 \int \frac {1}{-35+7 x} \, dx-\frac {8614904}{49} \int \frac {1}{-38+7 x} \, dx \\ & = 80 x-18 x^2+x^3-\frac {38}{7} \log (38-7 x)+\frac {1}{7} (38-7 x) \log \left (\frac {38-7 x}{5-x}\right )+\frac {38}{7} \log (5-x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=80 x-18 x^2+x^3-\frac {38}{7} \log (38-7 x)+\frac {38}{7} \log (5-x)-\frac {1}{7} (-38+7 x) \log \left (\frac {-38+7 x}{-5+x}\right ) \]
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Time = 2.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97
method | result | size |
norman | \(x^{3}+80 x -18 x^{2}-\ln \left (\frac {7 x -38}{-5+x}\right ) x\) | \(28\) |
risch | \(x^{3}+80 x -18 x^{2}-\ln \left (\frac {7 x -38}{-5+x}\right ) x\) | \(28\) |
parallelrisch | \(x^{3}+\frac {36186}{7}-18 x^{2}-\ln \left (\frac {7 x -38}{-5+x}\right ) x +80 x\) | \(29\) |
derivativedivides | \(-\frac {\ln \left (7-\frac {3}{-5+x}\right ) \left (7-\frac {3}{-5+x}\right ) \left (-5+x \right )}{7}+\left (-5+x \right )^{3}-3 \left (-5+x \right )^{2}+125-25 x -\frac {38 \ln \left (7-\frac {3}{-5+x}\right )}{7}\) | \(54\) |
default | \(-\frac {\ln \left (7-\frac {3}{-5+x}\right ) \left (7-\frac {3}{-5+x}\right ) \left (-5+x \right )}{7}+\left (-5+x \right )^{3}-3 \left (-5+x \right )^{2}+125-25 x -\frac {38 \ln \left (7-\frac {3}{-5+x}\right )}{7}\) | \(54\) |
parts | \(x^{3}-18 x^{2}+80 x +5 \ln \left (-5+x \right )-\frac {38 \ln \left (7 x -38\right )}{7}-\frac {3 \ln \left (-\frac {3}{-5+x}\right )}{7}-\frac {\ln \left (7-\frac {3}{-5+x}\right ) \left (7-\frac {3}{-5+x}\right ) \left (-5+x \right )}{7}\) | \(61\) |
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=x^{3} - 18 \, x^{2} - x \log \left (\frac {7 \, x - 38}{x - 5}\right ) + 80 \, x \]
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Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=x^{3} - 18 x^{2} - x \log {\left (\frac {7 x - 38}{x - 5} \right )} + 80 x \]
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (25) = 50\).
Time = 0.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.97 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=x^{3} - 18 \, x^{2} - \frac {1}{21} \, {\left (21 \, x + 1330 \, \log \left (x - 5\right ) - 114\right )} \log \left (7 \, x - 38\right ) + \frac {190}{3} \, \log \left (7 \, x - 38\right )^{2} + {\left (x - 5\right )} \log \left (x - 5\right ) - \frac {190}{3} \, \log \left (7 \, x - 38\right ) \log \left (x - 5\right ) + \frac {190}{3} \, \log \left (x - 5\right )^{2} - \frac {190}{3} \, {\left (\log \left (7 \, x - 38\right ) - \log \left (x - 5\right )\right )} \log \left (\frac {7 \, x}{x - 5} - \frac {38}{x - 5}\right ) + 80 \, x - \frac {38}{7} \, \log \left (7 \, x - 38\right ) + 5 \, \log \left (x - 5\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (25) = 50\).
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 4.03 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=\frac {3 \, {\left (\frac {25 \, {\left (7 \, x - 38\right )}^{2}}{{\left (x - 5\right )}^{2}} - \frac {359 \, {\left (7 \, x - 38\right )}}{x - 5} + 1279\right )}}{\frac {{\left (7 \, x - 38\right )}^{3}}{{\left (x - 5\right )}^{3}} - \frac {21 \, {\left (7 \, x - 38\right )}^{2}}{{\left (x - 5\right )}^{2}} + \frac {147 \, {\left (7 \, x - 38\right )}}{x - 5} - 343} + \frac {3 \, \log \left (\frac {7 \, x - 38}{x - 5}\right )}{\frac {7 \, x - 38}{x - 5} - 7} - 5 \, \log \left (\frac {7 \, x - 38}{x - 5}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=80\,x-x\,\ln \left (\frac {7\,x-38}{x-5}\right )-18\,x^2+x^3 \]
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