Optimal. Leaf size=78 \[ \frac{b^2}{a^3 n \left (a+b x^n\right )}-\frac{3 b^2 \log \left (a+b x^n\right )}{a^4 n}+\frac{3 b^2 \log (x)}{a^4}+\frac{2 b x^{-n}}{a^3 n}-\frac{x^{-2 n}}{2 a^2 n} \]
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Rubi [A] time = 0.0443638, antiderivative size = 78, 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} \[ \frac{b^2}{a^3 n \left (a+b x^n\right )}-\frac{3 b^2 \log \left (a+b x^n\right )}{a^4 n}+\frac{3 b^2 \log (x)}{a^4}+\frac{2 b x^{-n}}{a^3 n}-\frac{x^{-2 n}}{2 a^2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{x^{-1-2 n}}{\left (a+b x^n\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^3}-\frac{2 b}{a^3 x^2}+\frac{3 b^2}{a^4 x}-\frac{b^3}{a^3 (a+b x)^2}-\frac{3 b^3}{a^4 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-2 n}}{2 a^2 n}+\frac{2 b x^{-n}}{a^3 n}+\frac{b^2}{a^3 n \left (a+b x^n\right )}+\frac{3 b^2 \log (x)}{a^4}-\frac{3 b^2 \log \left (a+b x^n\right )}{a^4 n}\\ \end{align*}
Mathematica [A] time = 0.136932, size = 65, normalized size = 0.83 \[ \frac{a \left (\frac{2 b^2}{a+b x^n}-a x^{-2 n}+4 b x^{-n}\right )-6 b^2 \log \left (a+b x^n\right )+6 b^2 n \log (x)}{2 a^4 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 117, normalized size = 1.5 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2} \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) } \left ( -3\,{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}n}}-{\frac{1}{2\,an}}+{\frac{3\,b{{\rm e}^{n\ln \left ( x \right ) }}}{2\,{a}^{2}n}}+3\,{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}}+3\,{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}} \right ) }-3\,{\frac{{b}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{4}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02831, size = 230, normalized size = 2.95 \begin{align*} \frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) + 3 \, a^{2} b x^{n} - a^{3} + 6 \,{\left (a b^{2} n \log \left (x\right ) + a b^{2}\right )} x^{2 \, n} - 6 \,{\left (b^{3} x^{3 \, n} + a b^{2} x^{2 \, n}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (a^{4} b n x^{3 \, n} + a^{5} n x^{2 \, n}\right )}} \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}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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