Optimal. Leaf size=24 \[ -\frac{x^{-3 n} \left (a+b x^n\right )^3}{3 a n} \]
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Rubi [A] time = 0.0048236, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ -\frac{x^{-3 n} \left (a+b x^n\right )^3}{3 a n} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int x^{-1-3 n} \left (a+b x^n\right )^2 \, dx &=-\frac{x^{-3 n} \left (a+b x^n\right )^3}{3 a n}\\ \end{align*}
Mathematica [A] time = 0.0057434, size = 24, normalized size = 1. \[ -\frac{x^{-3 n} \left (a+b x^n\right )^3}{3 a n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 45, normalized size = 1.9 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ( -{\frac{{a}^{2}}{3\,n}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}-{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}b}{n}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24799, size = 70, normalized size = 2.92 \begin{align*} -\frac{3 \, b^{2} x^{2 \, n} + 3 \, a b x^{n} + a^{2}}{3 \, n x^{3 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 14.2528, size = 39, normalized size = 1.62 \begin{align*} \begin{cases} - \frac{a^{2} x^{- 3 n}}{3 n} - \frac{a b x^{- 2 n}}{n} - \frac{b^{2} x^{- n}}{n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{2} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22713, size = 45, normalized size = 1.88 \begin{align*} -\frac{3 \, b^{2} x^{2 \, n} + 3 \, a b x^{n} + a^{2}}{3 \, n x^{3 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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