Optimal. Leaf size=77 \[ -\frac{2 b}{a^3 n \left (a+b x^n\right )}-\frac{b}{2 a^2 n \left (a+b x^n\right )^2}+\frac{3 b \log \left (a+b x^n\right )}{a^4 n}-\frac{3 b \log (x)}{a^4}-\frac{x^{-n}}{a^3 n} \]
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Rubi [A] time = 0.0473548, antiderivative size = 77, 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} \[ -\frac{2 b}{a^3 n \left (a+b x^n\right )}-\frac{b}{2 a^2 n \left (a+b x^n\right )^2}+\frac{3 b \log \left (a+b x^n\right )}{a^4 n}-\frac{3 b \log (x)}{a^4}-\frac{x^{-n}}{a^3 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{x^{-1-n}}{\left (a+b x^n\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^3} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^2}-\frac{3 b}{a^4 x}+\frac{b^2}{a^2 (a+b x)^3}+\frac{2 b^2}{a^3 (a+b x)^2}+\frac{3 b^2}{a^4 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-n}}{a^3 n}-\frac{b}{2 a^2 n \left (a+b x^n\right )^2}-\frac{2 b}{a^3 n \left (a+b x^n\right )}-\frac{3 b \log (x)}{a^4}+\frac{3 b \log \left (a+b x^n\right )}{a^4 n}\\ \end{align*}
Mathematica [A] time = 0.196964, size = 58, normalized size = 0.75 \[ -\frac{\frac{a b \left (5 a+4 b x^n\right )}{\left (a+b x^n\right )^2}-6 b \log \left (a+b x^n\right )+2 a x^{-n}+6 b n \log (x)}{2 a^4 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 132, normalized size = 1.7 \begin{align*}{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }} \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ( -{\frac{1}{an}}-3\,{\frac{b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}}}-6\,{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}}+6\,{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-3\,{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}}+{\frac{9\,{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2\,{a}^{4}n}} \right ) }+3\,{\frac{b\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 [A] time = 0.97778, size = 123, normalized size = 1.6 \begin{align*} -\frac{6 \, b^{2} x^{2 \, n} + 9 \, a b x^{n} + 2 \, a^{2}}{2 \,{\left (a^{3} b^{2} n x^{3 \, n} + 2 \, a^{4} b n x^{2 \, n} + a^{5} n x^{n}\right )}} - \frac{3 \, b \log \left (x\right )}{a^{4}} + \frac{3 \, b \log \left (\frac{b x^{n} + a}{b}\right )}{a^{4} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.01348, size = 311, normalized size = 4.04 \begin{align*} -\frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) + 2 \, a^{3} + 6 \,{\left (2 \, a b^{2} n \log \left (x\right ) + a b^{2}\right )} x^{2 \, n} + 3 \,{\left (2 \, a^{2} b n \log \left (x\right ) + 3 \, a^{2} b\right )} x^{n} - 6 \,{\left (b^{3} x^{3 \, n} + 2 \, a b^{2} x^{2 \, n} + a^{2} b x^{n}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (a^{4} b^{2} n x^{3 \, n} + 2 \, a^{5} b n x^{2 \, n} + a^{6} n x^{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^{-n - 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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