Optimal. Leaf size=76 \[ -\frac{b^2 x^{-n}}{a^3 n}+\frac{b^3 \log \left (a+b x^n\right )}{a^4 n}-\frac{b^3 \log (x)}{a^4}+\frac{b x^{-2 n}}{2 a^2 n}-\frac{x^{-3 n}}{3 a n} \]
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Rubi [A] time = 0.0360553, antiderivative size = 76, 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} \[ -\frac{b^2 x^{-n}}{a^3 n}+\frac{b^3 \log \left (a+b x^n\right )}{a^4 n}-\frac{b^3 \log (x)}{a^4}+\frac{b x^{-2 n}}{2 a^2 n}-\frac{x^{-3 n}}{3 a n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{x^{-1-3 n}}{a+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^4}-\frac{b}{a^2 x^3}+\frac{b^2}{a^3 x^2}-\frac{b^3}{a^4 x}+\frac{b^4}{a^4 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-3 n}}{3 a n}+\frac{b x^{-2 n}}{2 a^2 n}-\frac{b^2 x^{-n}}{a^3 n}-\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log \left (a+b x^n\right )}{a^4 n}\\ \end{align*}
Mathematica [A] time = 0.0738045, size = 62, normalized size = 0.82 \[ -\frac{a x^{-3 n} \left (2 a^2-3 a b x^n+6 b^2 x^{2 n}\right )-6 b^3 \log \left (a+b x^n\right )+6 b^3 n \log (x)}{6 a^4 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 88, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ( -{\frac{1}{3\,an}}+{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{2\,{a}^{2}n}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}} \right ) }+{\frac{{b}^{3}\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 = 0.975223, size = 96, normalized size = 1.26 \begin{align*} -\frac{b^{3} \log \left (x\right )}{a^{4}} + \frac{b^{3} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{4} n} - \frac{6 \, b^{2} x^{2 \, n} - 3 \, a b x^{n} + 2 \, a^{2}}{6 \, a^{3} n x^{3 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.01885, size = 159, normalized size = 2.09 \begin{align*} -\frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) - 6 \, b^{3} x^{3 \, n} \log \left (b x^{n} + a\right ) + 6 \, a b^{2} x^{2 \, n} - 3 \, a^{2} b x^{n} + 2 \, a^{3}}{6 \, a^{4} n x^{3 \, 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^{-3 \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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