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 = 0.0097024, antiderivative size = 22, normalized size of antiderivative = 1., 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 1593
Rule 266
Rule 36
Rule 29
Rule 31
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*}
Mathematica [A] time = 0.0059449, size = 22, normalized size = 1. \[ \frac{n \log (x)-\log \left (3 x^n+2\right )}{2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 32, normalized size = 1.5 \begin{align*}{\frac{\ln \left ( x \right ) }{2\,n}}+{\frac{\ln \left ( x \right ) }{2}}-{\frac{\ln \left ( 2\,x+3\,{{\rm e}^{ \left ( 1+n \right ) \ln \left ( x \right ) }} \right ) }{2\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11082, size = 22, normalized size = 1. \begin{align*} -\frac{\log \left (x^{n} + \frac{2}{3}\right )}{2 \, n} + \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.831635, size = 66, normalized size = 3. \begin{align*} \frac{{\left (n + 1\right )} \log \left (x\right ) - \log \left (3 \, x^{n + 1} + 2 \, x\right )}{2 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.989, size = 20, normalized size = 0.91 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{2} - \frac{\log{\left (x^{n} + \frac{2}{3} \right )}}{2 n} & \text{for}\: n \neq 0 \\\frac{\log{\left (x \right )}}{5} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{3 \, x^{n + 1} + 2 \, x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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