Integrand size = 13, antiderivative size = 22 \[ \int \frac {1}{2 x+3 x^{1+n}} \, dx=\frac {\log (x)}{2}-\frac {\log \left (2+3 x^n\right )}{2 n} \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1607, 272, 36, 29, 31} \[ \int \frac {1}{2 x+3 x^{1+n}} \, dx=\frac {\log (x)}{2}-\frac {\log \left (3 x^n+2\right )}{2 n} \]
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (2+3 x^n\right )} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {1}{x (2+3 x)} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{2 n}-\frac {3 \text {Subst}\left (\int \frac {1}{2+3 x} \, dx,x,x^n\right )}{2 n} \\ & = \frac {\log (x)}{2}-\frac {\log \left (2+3 x^n\right )}{2 n} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{2 x+3 x^{1+n}} \, dx=\frac {\log \left (x^n\right )-\log \left (n \left (2+3 x^n\right )\right )}{2 n} \]
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Time = 1.81 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(\frac {n \ln \left (x \right )+\ln \left (x \right )-\ln \left (x +\frac {3 x^{1+n}}{2}\right )}{2 n}\) | \(25\) |
meijerg | \(\frac {n \ln \left (x \right )+\ln \left (3\right )-\ln \left (2\right )-\ln \left (1+\frac {3 x^{n}}{2}\right )}{2 n}\) | \(27\) |
risch | \(\frac {\ln \left (x \right )}{2 n}+\frac {\ln \left (x \right )}{2}-\frac {\ln \left (\frac {2 x}{3}+x^{1+n}\right )}{2 n}\) | \(28\) |
norman | \(\frac {\left (1+n \right ) \ln \left (x \right )}{2 n}-\frac {\ln \left (2 x +3 \,{\mathrm e}^{\left (1+n \right ) \ln \left (x \right )}\right )}{2 n}\) | \(31\) |
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Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {1}{2 x+3 x^{1+n}} \, dx=\frac {{\left (n + 1\right )} \log \left (x\right ) - \log \left (3 \, x^{n + 1} + 2 \, x\right )}{2 \, n} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {1}{2 x+3 x^{1+n}} \, dx=\begin {cases} \frac {\log {\left (x \right )}}{2} + \frac {\log {\left (x \right )}}{2 n} - \frac {\log {\left (2 x + 3 x^{n + 1} \right )}}{2 n} & \text {for}\: n \neq 0 \\\frac {\log {\left (x \right )}}{5} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {1}{2 x+3 x^{1+n}} \, dx=-\frac {\log \left (x^{n} + \frac {2}{3}\right )}{2 \, n} + \frac {1}{2} \, \log \left (x\right ) \]
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\[ \int \frac {1}{2 x+3 x^{1+n}} \, dx=\int { \frac {1}{3 \, x^{n + 1} + 2 \, x} \,d x } \]
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Time = 9.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {1}{2 x+3 x^{1+n}} \, dx=\frac {\ln \left (x\right )\,\left (n+1\right )}{2\,n}-\frac {\ln \left (\frac {2\,x}{3}+x^{n+1}\right )}{2\,n} \]
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