Optimal. Leaf size=28 \[ \frac{x^n}{b n}-\frac{a \log \left (a+b x^n\right )}{b^2 n} \]
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Rubi [A] time = 0.0172597, antiderivative size = 28, 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} \[ \frac{x^n}{b n}-\frac{a \log \left (a+b x^n\right )}{b^2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{-1+2 n}}{a+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{a+b x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{x^n}{b n}-\frac{a \log \left (a+b x^n\right )}{b^2 n}\\ \end{align*}
Mathematica [A] time = 0.0125486, size = 26, normalized size = 0.93 \[ \frac{\frac{x^n}{b}-\frac{a \log \left (a+b x^n\right )}{b^2}}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 33, normalized size = 1.2 \begin{align*}{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{bn}}-{\frac{a\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{2}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971104, size = 43, normalized size = 1.54 \begin{align*} \frac{x^{n}}{b n} - \frac{a \log \left (\frac{b x^{n} + a}{b}\right )}{b^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07346, size = 49, normalized size = 1.75 \begin{align*} \frac{b x^{n} - a \log \left (b x^{n} + a\right )}{b^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.1807, size = 41, normalized size = 1.46 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 0 \\\frac{x^{2 n}}{2 a n} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\- \frac{a \log{\left (\frac{a}{b} + x^{n} \right )}}{b^{2} n} + \frac{x^{n}}{b 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^{2 \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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