Optimal. Leaf size=45 \[ -\frac{a^2 x^{-6 n}}{6 n}-\frac{2 a b x^{-5 n}}{5 n}-\frac{b^2 x^{-4 n}}{4 n} \]
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Rubi [A] time = 0.0170153, antiderivative size = 45, 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} \[ -\frac{a^2 x^{-6 n}}{6 n}-\frac{2 a b x^{-5 n}}{5 n}-\frac{b^2 x^{-4 n}}{4 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-6 n} \left (a+b x^n\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^7} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^7}+\frac{2 a b}{x^6}+\frac{b^2}{x^5}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^2 x^{-6 n}}{6 n}-\frac{2 a b x^{-5 n}}{5 n}-\frac{b^2 x^{-4 n}}{4 n}\\ \end{align*}
Mathematica [A] time = 0.015463, size = 35, normalized size = 0.78 \[ -\frac{x^{-6 n} \left (10 a^2+24 a b x^n+15 b^2 x^{2 n}\right )}{60 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 40, normalized size = 0.9 \begin{align*} -{\frac{{b}^{2}}{4\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{2\,ab}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{{a}^{2}}{6\,n \left ({x}^{n} \right ) ^{6}}} \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 = 1.33876, size = 78, normalized size = 1.73 \begin{align*} -\frac{15 \, b^{2} x^{2 \, n} + 24 \, a b x^{n} + 10 \, a^{2}}{60 \, n x^{6 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.3841, size = 46, normalized size = 1.02 \begin{align*} \begin{cases} - \frac{a^{2} x^{- 6 n}}{6 n} - \frac{2 a b x^{- 5 n}}{5 n} - \frac{b^{2} x^{- 4 n}}{4 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 = 1.16786, size = 47, normalized size = 1.04 \begin{align*} -\frac{15 \, b^{2} x^{2 \, n} + 24 \, a b x^{n} + 10 \, a^{2}}{60 \, n x^{6 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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