Integrand size = 13, antiderivative size = 28 \[ \int \frac {1}{c x^2+d x^3} \, dx=-\frac {1}{c x}-\frac {d \log (x)}{c^2}+\frac {d \log (c+d x)}{c^2} \]
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Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1607, 46} \[ \int \frac {1}{c x^2+d x^3} \, dx=-\frac {d \log (x)}{c^2}+\frac {d \log (c+d x)}{c^2}-\frac {1}{c x} \]
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Rule 46
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 (c+d x)} \, dx \\ & = \int \left (\frac {1}{c x^2}-\frac {d}{c^2 x}+\frac {d^2}{c^2 (c+d x)}\right ) \, dx \\ & = -\frac {1}{c x}-\frac {d \log (x)}{c^2}+\frac {d \log (c+d x)}{c^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c x^2+d x^3} \, dx=-\frac {1}{c x}-\frac {d \log (x)}{c^2}+\frac {d \log (c+d x)}{c^2} \]
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Time = 0.69 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
method | result | size |
parallelrisch | \(-\frac {d \ln \left (x \right ) x -d \ln \left (d x +c \right ) x +c}{c^{2} x}\) | \(26\) |
default | \(-\frac {1}{c x}-\frac {d \ln \left (x \right )}{c^{2}}+\frac {d \ln \left (d x +c \right )}{c^{2}}\) | \(29\) |
norman | \(-\frac {1}{c x}-\frac {d \ln \left (x \right )}{c^{2}}+\frac {d \ln \left (d x +c \right )}{c^{2}}\) | \(29\) |
risch | \(-\frac {1}{c x}+\frac {d \ln \left (-d x -c \right )}{c^{2}}-\frac {d \ln \left (x \right )}{c^{2}}\) | \(32\) |
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Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {1}{c x^2+d x^3} \, dx=\frac {d x \log \left (d x + c\right ) - d x \log \left (x\right ) - c}{c^{2} x} \]
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Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {1}{c x^2+d x^3} \, dx=- \frac {1}{c x} + \frac {d \left (- \log {\left (x \right )} + \log {\left (\frac {c}{d} + x \right )}\right )}{c^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c x^2+d x^3} \, dx=\frac {d \log \left (d x + c\right )}{c^{2}} - \frac {d \log \left (x\right )}{c^{2}} - \frac {1}{c x} \]
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Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{c x^2+d x^3} \, dx=\frac {d \log \left ({\left | d x + c \right |}\right )}{c^{2}} - \frac {d \log \left ({\left | x \right |}\right )}{c^{2}} - \frac {1}{c x} \]
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Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {1}{c x^2+d x^3} \, dx=\frac {2\,d\,\mathrm {atanh}\left (\frac {2\,d\,x}{c}+1\right )}{c^2}-\frac {1}{c\,x} \]
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