Optimal. Leaf size=94 \[ -\frac{3 b^2 x^{-n}}{a^4 n}-\frac{b^3}{a^4 n \left (a+b x^n\right )}+\frac{4 b^3 \log \left (a+b x^n\right )}{a^5 n}-\frac{4 b^3 \log (x)}{a^5}+\frac{b x^{-2 n}}{a^3 n}-\frac{x^{-3 n}}{3 a^2 n} \]
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Rubi [A] time = 0.0560242, antiderivative size = 94, 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} \[ -\frac{3 b^2 x^{-n}}{a^4 n}-\frac{b^3}{a^4 n \left (a+b x^n\right )}+\frac{4 b^3 \log \left (a+b x^n\right )}{a^5 n}-\frac{4 b^3 \log (x)}{a^5}+\frac{b x^{-2 n}}{a^3 n}-\frac{x^{-3 n}}{3 a^2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{x^{-1-3 n}}{\left (a+b x^n\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^4}-\frac{2 b}{a^3 x^3}+\frac{3 b^2}{a^4 x^2}-\frac{4 b^3}{a^5 x}+\frac{b^4}{a^4 (a+b x)^2}+\frac{4 b^4}{a^5 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-3 n}}{3 a^2 n}+\frac{b x^{-2 n}}{a^3 n}-\frac{3 b^2 x^{-n}}{a^4 n}-\frac{b^3}{a^4 n \left (a+b x^n\right )}-\frac{4 b^3 \log (x)}{a^5}+\frac{4 b^3 \log \left (a+b x^n\right )}{a^5 n}\\ \end{align*}
Mathematica [A] time = 0.146716, size = 78, normalized size = 0.83 \[ \frac{a \left (-a^2 x^{-3 n}-\frac{3 b^3}{a+b x^n}+3 a b x^{-2 n}-9 b^2 x^{-n}\right )+12 b^3 \log \left (a+b x^n\right )-12 b^3 n \log (x)}{3 a^5 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 135, normalized size = 1.4 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3} \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) } \left ( 4\,{\frac{{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{{a}^{5}n}}-{\frac{1}{3\,an}}+{\frac{2\,b{{\rm e}^{n\ln \left ( x \right ) }}}{3\,{a}^{2}n}}-2\,{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-4\,{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}}-4\,{\frac{{b}^{4}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{{a}^{5}}} \right ) }+4\,{\frac{{b}^{3}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{5}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969758, size = 127, normalized size = 1.35 \begin{align*} -\frac{12 \, b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} - 2 \, a^{2} b x^{n} + a^{3}}{3 \,{\left (a^{4} b n x^{4 \, n} + a^{5} n x^{3 \, n}\right )}} - \frac{4 \, b^{3} \log \left (x\right )}{a^{5}} + \frac{4 \, b^{3} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{5} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07984, size = 262, normalized size = 2.79 \begin{align*} -\frac{12 \, b^{4} n x^{4 \, n} \log \left (x\right ) + 6 \, a^{2} b^{2} x^{2 \, n} - 2 \, a^{3} b x^{n} + a^{4} + 12 \,{\left (a b^{3} n \log \left (x\right ) + a b^{3}\right )} x^{3 \, n} - 12 \,{\left (b^{4} x^{4 \, n} + a b^{3} x^{3 \, n}\right )} \log \left (b x^{n} + a\right )}{3 \,{\left (a^{5} b n x^{4 \, n} + a^{6} n x^{3 \, 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^{-3 \, 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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