Integrand size = 16, antiderivative size = 33 \[ \int \frac {x}{15+\frac {2}{x^2}+\frac {13}{x}} \, dx=-\frac {13 x}{225}+\frac {x^2}{30}+\frac {8}{189} \log (2+3 x)-\frac {1}{875} \log (1+5 x) \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1368, 715, 646, 31} \[ \int \frac {x}{15+\frac {2}{x^2}+\frac {13}{x}} \, dx=\frac {x^2}{30}-\frac {13 x}{225}+\frac {8}{189} \log (3 x+2)-\frac {1}{875} \log (5 x+1) \]
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Rule 31
Rule 646
Rule 715
Rule 1368
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{2+13 x+15 x^2} \, dx \\ & = \int \left (-\frac {13}{225}+\frac {x}{15}+\frac {26+139 x}{225 \left (2+13 x+15 x^2\right )}\right ) \, dx \\ & = -\frac {13 x}{225}+\frac {x^2}{30}+\frac {1}{225} \int \frac {26+139 x}{2+13 x+15 x^2} \, dx \\ & = -\frac {13 x}{225}+\frac {x^2}{30}-\frac {3}{175} \int \frac {1}{3+15 x} \, dx+\frac {40}{63} \int \frac {1}{10+15 x} \, dx \\ & = -\frac {13 x}{225}+\frac {x^2}{30}+\frac {8}{189} \log (2+3 x)-\frac {1}{875} \log (1+5 x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {x}{15+\frac {2}{x^2}+\frac {13}{x}} \, dx=-\frac {13 x}{225}+\frac {x^2}{30}+\frac {8}{189} \log (2+3 x)-\frac {1}{875} \log (1+5 x) \]
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Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67
method | result | size |
parallelrisch | \(\frac {x^{2}}{30}-\frac {13 x}{225}-\frac {\ln \left (x +\frac {1}{5}\right )}{875}+\frac {8 \ln \left (x +\frac {2}{3}\right )}{189}\) | \(22\) |
default | \(-\frac {13 x}{225}+\frac {x^{2}}{30}+\frac {8 \ln \left (3 x +2\right )}{189}-\frac {\ln \left (1+5 x \right )}{875}\) | \(26\) |
norman | \(-\frac {13 x}{225}+\frac {x^{2}}{30}+\frac {8 \ln \left (3 x +2\right )}{189}-\frac {\ln \left (1+5 x \right )}{875}\) | \(26\) |
risch | \(-\frac {13 x}{225}+\frac {x^{2}}{30}+\frac {8 \ln \left (3 x +2\right )}{189}-\frac {\ln \left (1+5 x \right )}{875}\) | \(26\) |
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Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {x}{15+\frac {2}{x^2}+\frac {13}{x}} \, dx=\frac {1}{30} \, x^{2} - \frac {13}{225} \, x - \frac {1}{875} \, \log \left (5 \, x + 1\right ) + \frac {8}{189} \, \log \left (3 \, x + 2\right ) \]
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Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {x}{15+\frac {2}{x^2}+\frac {13}{x}} \, dx=\frac {x^{2}}{30} - \frac {13 x}{225} - \frac {\log {\left (x + \frac {1}{5} \right )}}{875} + \frac {8 \log {\left (x + \frac {2}{3} \right )}}{189} \]
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Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {x}{15+\frac {2}{x^2}+\frac {13}{x}} \, dx=\frac {1}{30} \, x^{2} - \frac {13}{225} \, x - \frac {1}{875} \, \log \left (5 \, x + 1\right ) + \frac {8}{189} \, \log \left (3 \, x + 2\right ) \]
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Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {x}{15+\frac {2}{x^2}+\frac {13}{x}} \, dx=\frac {1}{30} \, x^{2} - \frac {13}{225} \, x - \frac {1}{875} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) + \frac {8}{189} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]
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Time = 8.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.64 \[ \int \frac {x}{15+\frac {2}{x^2}+\frac {13}{x}} \, dx=\frac {8\,\ln \left (x+\frac {2}{3}\right )}{189}-\frac {13\,x}{225}-\frac {\ln \left (x+\frac {1}{5}\right )}{875}+\frac {x^2}{30} \]
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