Optimal. Leaf size=19 \[ -\frac{1}{2 b n \left (a+b x^n\right )^2} \]
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Rubi [A] time = 0.0045467, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ -\frac{1}{2 b n \left (a+b x^n\right )^2} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int \frac{x^{-1+n}}{\left (a+b x^n\right )^3} \, dx &=-\frac{1}{2 b n \left (a+b x^n\right )^2}\\ \end{align*}
Mathematica [A] time = 0.004409, size = 19, normalized size = 1. \[ -\frac{1}{2 b n \left (a+b x^n\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 20, normalized size = 1.1 \begin{align*} -{\frac{1}{2\,bn \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982769, size = 23, normalized size = 1.21 \begin{align*} -\frac{1}{2 \,{\left (b x^{n} + a\right )}^{2} b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.00794, size = 65, normalized size = 3.42 \begin{align*} -\frac{1}{2 \,{\left (b^{3} n x^{2 \, n} + 2 \, a b^{2} n x^{n} + a^{2} b n\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 22.8239, size = 109, normalized size = 5.74 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{b^{3}} & \text{for}\: a = 0 \wedge n = 0 \\- \frac{x^{- 2 n}}{2 b^{3} n} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{\left (a + b\right )^{3}} & \text{for}\: n = 0 \\\frac{2 a x^{n}}{2 a^{4} n + 4 a^{3} b n x^{n} + 2 a^{2} b^{2} n x^{2 n}} + \frac{b x^{2 n}}{2 a^{4} n + 4 a^{3} b n x^{n} + 2 a^{2} b^{2} n x^{2 n}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12032, size = 23, normalized size = 1.21 \begin{align*} -\frac{1}{2 \,{\left (b x^{n} + a\right )}^{2} b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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