Optimal. Leaf size=22 \[ \frac {\log (x)}{2}-\frac {\log \left (3 x^n+2\right )}{2 n} \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1593, 266, 36, 29, 31} \[ \frac {\log (x)}{2}-\frac {\log \left (3 x^n+2\right )}{2 n} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{2 x+3 x^{1+n}} \, dx &=\int \frac {1}{x \left (2+3 x^n\right )} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (2+3 x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{2 n}-\frac {3 \operatorname {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*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ \frac {n \log (x)-\log \left (3 x^n+2\right )}{2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 26, normalized size = 1.18 \[ \frac {{\left (n + 1\right )} \log \relax (x) - \log \left (3 \, x^{n + 1} + 2 \, x\right )}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{3 \, x^{n + 1} + 2 \, x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 1.45 \[ \frac {\ln \relax (x )}{2}+\frac {\ln \relax (x )}{2 n}-\frac {\ln \left (2 x +3 \,{\mathrm e}^{\left (n +1\right ) \ln \relax (x )}\right )}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 16, normalized size = 0.73 \[ -\frac {\log \left (x^{n} + \frac {2}{3}\right )}{2 \, n} + \frac {1}{2} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.23, size = 26, normalized size = 1.18 \[ \frac {\ln \relax (x)\,\left (n+1\right )}{2\,n}-\frac {\ln \left (\frac {2\,x}{3}+x^{n+1}\right )}{2\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.51, size = 20, normalized size = 0.91 \[ \begin {cases} \frac {\log {\relax (x )}}{2} - \frac {\log {\left (x^{n} + \frac {2}{3} \right )}}{2 n} & \text {for}\: n \neq 0 \\\frac {\log {\relax (x )}}{5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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