Integrand size = 16, antiderivative size = 40 \[ \int \frac {x^3}{13+\frac {2}{x}+15 x} \, dx=\frac {139 x}{3375}-\frac {13 x^2}{450}+\frac {x^3}{45}-\frac {16}{567} \log (2+3 x)+\frac {\log (1+5 x)}{4375} \]
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Time = 0.01 (sec) , antiderivative size = 40, 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 = {1400, 715, 646, 31} \[ \int \frac {x^3}{13+\frac {2}{x}+15 x} \, dx=\frac {x^3}{45}-\frac {13 x^2}{450}+\frac {139 x}{3375}-\frac {16}{567} \log (3 x+2)+\frac {\log (5 x+1)}{4375} \]
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Rule 31
Rule 646
Rule 715
Rule 1400
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4}{2+13 x+15 x^2} \, dx \\ & = \int \left (\frac {139}{3375}-\frac {13 x}{225}+\frac {x^2}{15}-\frac {278+1417 x}{3375 \left (2+13 x+15 x^2\right )}\right ) \, dx \\ & = \frac {139 x}{3375}-\frac {13 x^2}{450}+\frac {x^3}{45}-\frac {\int \frac {278+1417 x}{2+13 x+15 x^2} \, dx}{3375} \\ & = \frac {139 x}{3375}-\frac {13 x^2}{450}+\frac {x^3}{45}+\frac {3}{875} \int \frac {1}{3+15 x} \, dx-\frac {80}{189} \int \frac {1}{10+15 x} \, dx \\ & = \frac {139 x}{3375}-\frac {13 x^2}{450}+\frac {x^3}{45}-\frac {16}{567} \log (2+3 x)+\frac {\log (1+5 x)}{4375} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{13+\frac {2}{x}+15 x} \, dx=\frac {139 x}{3375}-\frac {13 x^2}{450}+\frac {x^3}{45}-\frac {16}{567} \log (2+3 x)+\frac {\log (1+5 x)}{4375} \]
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Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.68
method | result | size |
parallelrisch | \(\frac {x^{3}}{45}-\frac {13 x^{2}}{450}+\frac {139 x}{3375}+\frac {\ln \left (x +\frac {1}{5}\right )}{4375}-\frac {16 \ln \left (x +\frac {2}{3}\right )}{567}\) | \(27\) |
default | \(\frac {139 x}{3375}-\frac {13 x^{2}}{450}+\frac {x^{3}}{45}-\frac {16 \ln \left (3 x +2\right )}{567}+\frac {\ln \left (1+5 x \right )}{4375}\) | \(31\) |
norman | \(\frac {139 x}{3375}-\frac {13 x^{2}}{450}+\frac {x^{3}}{45}-\frac {16 \ln \left (3 x +2\right )}{567}+\frac {\ln \left (1+5 x \right )}{4375}\) | \(31\) |
risch | \(\frac {139 x}{3375}-\frac {13 x^{2}}{450}+\frac {x^{3}}{45}-\frac {16 \ln \left (3 x +2\right )}{567}+\frac {\ln \left (1+5 x \right )}{4375}\) | \(31\) |
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Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.75 \[ \int \frac {x^3}{13+\frac {2}{x}+15 x} \, dx=\frac {1}{45} \, x^{3} - \frac {13}{450} \, x^{2} + \frac {139}{3375} \, x + \frac {1}{4375} \, \log \left (5 \, x + 1\right ) - \frac {16}{567} \, \log \left (3 \, x + 2\right ) \]
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Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{13+\frac {2}{x}+15 x} \, dx=\frac {x^{3}}{45} - \frac {13 x^{2}}{450} + \frac {139 x}{3375} + \frac {\log {\left (x + \frac {1}{5} \right )}}{4375} - \frac {16 \log {\left (x + \frac {2}{3} \right )}}{567} \]
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Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.75 \[ \int \frac {x^3}{13+\frac {2}{x}+15 x} \, dx=\frac {1}{45} \, x^{3} - \frac {13}{450} \, x^{2} + \frac {139}{3375} \, x + \frac {1}{4375} \, \log \left (5 \, x + 1\right ) - \frac {16}{567} \, \log \left (3 \, x + 2\right ) \]
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Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.80 \[ \int \frac {x^3}{13+\frac {2}{x}+15 x} \, dx=\frac {1}{45} \, x^{3} - \frac {13}{450} \, x^{2} + \frac {139}{3375} \, x + \frac {1}{4375} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) - \frac {16}{567} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]
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Time = 9.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.65 \[ \int \frac {x^3}{13+\frac {2}{x}+15 x} \, dx=\frac {139\,x}{3375}-\frac {16\,\ln \left (x+\frac {2}{3}\right )}{567}+\frac {\ln \left (x+\frac {1}{5}\right )}{4375}-\frac {13\,x^2}{450}+\frac {x^3}{45} \]
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