Optimal. Leaf size=57 \[ -\frac{b^2 \log \left (a+b x^n\right )}{a^3 n}+\frac{b^2 \log (x)}{a^3}+\frac{b x^{-n}}{a^2 n}-\frac{x^{-2 n}}{2 a n} \]
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Rubi [A] time = 0.0293391, antiderivative size = 57, 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, 44} \[ -\frac{b^2 \log \left (a+b x^n\right )}{a^3 n}+\frac{b^2 \log (x)}{a^3}+\frac{b x^{-n}}{a^2 n}-\frac{x^{-2 n}}{2 a n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{x^{-1-2 n}}{a+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^3}-\frac{b}{a^2 x^2}+\frac{b^2}{a^3 x}-\frac{b^3}{a^3 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-2 n}}{2 a n}+\frac{b x^{-n}}{a^2 n}+\frac{b^2 \log (x)}{a^3}-\frac{b^2 \log \left (a+b x^n\right )}{a^3 n}\\ \end{align*}
Mathematica [A] time = 0.0542028, size = 49, normalized size = 0.86 \[ \frac{-2 b^2 \log \left (a+b x^n\right )+a x^{-2 n} \left (2 b x^n-a\right )+2 b^2 n \log (x)}{2 a^3 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 69, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}n}}-{\frac{1}{2\,an}}+{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}} \right ) }-{\frac{{b}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{3}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964899, size = 78, normalized size = 1.37 \begin{align*} \frac{b^{2} \log \left (x\right )}{a^{3}} - \frac{b^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{3} n} + \frac{2 \, b x^{n} - a}{2 \, a^{2} n x^{2 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.04845, size = 128, normalized size = 2.25 \begin{align*} \frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) - 2 \, b^{2} x^{2 \, n} \log \left (b x^{n} + a\right ) + 2 \, a b x^{n} - a^{2}}{2 \, a^{3} n x^{2 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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