Optimal. Leaf size=43 \[ -\frac{2 x^n}{b^2 n}+\frac{4 \log \left (b x^n+2\right )}{b^3 n}+\frac{x^{2 n}}{2 b n} \]
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Rubi [A] time = 0.0214277, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ -\frac{2 x^n}{b^2 n}+\frac{4 \log \left (b x^n+2\right )}{b^3 n}+\frac{x^{2 n}}{2 b n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{-1+3 n}}{2+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{2+b x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{2}{b^2}+\frac{x}{b}+\frac{4}{b^2 (2+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{2 x^n}{b^2 n}+\frac{x^{2 n}}{2 b n}+\frac{4 \log \left (2+b x^n\right )}{b^3 n}\\ \end{align*}
Mathematica [A] time = 0.0168301, size = 33, normalized size = 0.77 \[ \frac{b x^n \left (b x^n-4\right )+8 \log \left (b x^n+2\right )}{2 b^3 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 48, normalized size = 1.1 \begin{align*} -2\,{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{2}n}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,bn}}+4\,{\frac{\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{3}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980038, size = 57, normalized size = 1.33 \begin{align*} \frac{b x^{2 \, n} - 4 \, x^{n}}{2 \, b^{2} n} + \frac{4 \, \log \left (\frac{b x^{n} + 2}{b}\right )}{b^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06841, size = 76, normalized size = 1.77 \begin{align*} \frac{b^{2} x^{2 \, n} - 4 \, b x^{n} + 8 \, \log \left (b x^{n} + 2\right )}{2 \, b^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 21.6661, size = 53, normalized size = 1.23 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{2} & \text{for}\: b = 0 \wedge n = 0 \\\frac{x^{3 n}}{6 n} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{b + 2} & \text{for}\: n = 0 \\\frac{x^{2 n}}{2 b n} - \frac{2 x^{n}}{b^{2} n} + \frac{4 \log{\left (x^{n} + \frac{2}{b} \right )}}{b^{3} n} & \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{x^{3 \, n - 1}}{b x^{n} + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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