Optimal. Leaf size=56 \[ -\frac{a^2}{2 b^3 n \left (a+b x^n\right )^2}+\frac{2 a}{b^3 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^3 n} \]
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Rubi [A] time = 0.0330859, antiderivative size = 56, 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, 43} \[ -\frac{a^2}{2 b^3 n \left (a+b x^n\right )^2}+\frac{2 a}{b^3 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^3 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{-1+3 n}}{\left (a+b x^n\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^3} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^3}-\frac{2 a}{b^2 (a+b x)^2}+\frac{1}{b^2 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^2}{2 b^3 n \left (a+b x^n\right )^2}+\frac{2 a}{b^3 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^3 n}\\ \end{align*}
Mathematica [A] time = 0.0386791, size = 42, normalized size = 0.75 \[ \frac{\frac{a \left (3 a+4 b x^n\right )}{\left (a+b x^n\right )^2}+2 \log \left (a+b x^n\right )}{2 b^3 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 57, normalized size = 1. \begin{align*}{\frac{1}{ \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{3\,{a}^{2}}{2\,{b}^{3}n}}+2\,{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{2}n}} \right ) }+{\frac{\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{3}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966206, size = 89, normalized size = 1.59 \begin{align*} \frac{4 \, a b x^{n} + 3 \, a^{2}}{2 \,{\left (b^{5} n x^{2 \, n} + 2 \, a b^{4} n x^{n} + a^{2} b^{3} n\right )}} + \frac{\log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.04072, size = 162, normalized size = 2.89 \begin{align*} \frac{4 \, a b x^{n} + 3 \, a^{2} + 2 \,{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (b^{5} n x^{2 \, n} + 2 \, a b^{4} n x^{n} + a^{2} b^{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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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