Integrand size = 10, antiderivative size = 15 \[ \int \frac {1+x+x^3}{x^2} \, dx=-\frac {1}{x}+\frac {x^2}{2}+\log (x) \]
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Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {14} \[ \int \frac {1+x+x^3}{x^2} \, dx=\frac {x^2}{2}-\frac {1}{x}+\log (x) \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x^2}+\frac {1}{x}+x\right ) \, dx \\ & = -\frac {1}{x}+\frac {x^2}{2}+\log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1+x+x^3}{x^2} \, dx=-\frac {1}{x}+\frac {x^2}{2}+\log (x) \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {1}{x}+\frac {x^{2}}{2}+\ln \left (x \right )\) | \(14\) |
risch | \(-\frac {1}{x}+\frac {x^{2}}{2}+\ln \left (x \right )\) | \(14\) |
norman | \(\frac {-1+\frac {x^{3}}{2}}{x}+\ln \left (x \right )\) | \(15\) |
parallelrisch | \(\frac {x^{3}+2 \ln \left (x \right ) x -2}{2 x}\) | \(16\) |
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none
Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1+x+x^3}{x^2} \, dx=\frac {x^{3} + 2 \, x \log \left (x\right ) - 2}{2 \, x} \]
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Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {1+x+x^3}{x^2} \, dx=\frac {x^{2}}{2} + \log {\left (x \right )} - \frac {1}{x} \]
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none
Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1+x+x^3}{x^2} \, dx=\frac {1}{2} \, x^{2} - \frac {1}{x} + \log \left (x\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {1+x+x^3}{x^2} \, dx=\frac {1}{2} \, x^{2} - \frac {1}{x} + \log \left ({\left | x \right |}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1+x+x^3}{x^2} \, dx=\ln \left (x\right )-\frac {1}{x}+\frac {x^2}{2} \]
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