Optimal. Leaf size=63 \[ -\frac{3 a^2 b x^{-5 n}}{5 n}-\frac{a^3 x^{-6 n}}{6 n}-\frac{3 a b^2 x^{-4 n}}{4 n}-\frac{b^3 x^{-3 n}}{3 n} \]
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Rubi [A] time = 0.0225921, antiderivative size = 63, 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} \[ -\frac{3 a^2 b x^{-5 n}}{5 n}-\frac{a^3 x^{-6 n}}{6 n}-\frac{3 a b^2 x^{-4 n}}{4 n}-\frac{b^3 x^{-3 n}}{3 n} \]
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 )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^7} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^3}{x^7}+\frac{3 a^2 b}{x^6}+\frac{3 a b^2}{x^5}+\frac{b^3}{x^4}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^3 x^{-6 n}}{6 n}-\frac{3 a^2 b x^{-5 n}}{5 n}-\frac{3 a b^2 x^{-4 n}}{4 n}-\frac{b^3 x^{-3 n}}{3 n}\\ \end{align*}
Mathematica [A] time = 0.0195597, size = 48, normalized size = 0.76 \[ -\frac{x^{-6 n} \left (36 a^2 b x^n+10 a^3+45 a b^2 x^{2 n}+20 b^3 x^{3 n}\right )}{60 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 56, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{3\,{b}^{2}a}{4\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{3\,b{a}^{2}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{{a}^{3}}{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 = 1.30937, size = 107, normalized size = 1.7 \begin{align*} -\frac{20 \, b^{3} x^{3 \, n} + 45 \, a b^{2} x^{2 \, n} + 36 \, a^{2} b x^{n} + 10 \, a^{3}}{60 \, n x^{6 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 40.1295, size = 63, normalized size = 1. \begin{align*} \begin{cases} - \frac{a^{3} x^{- 6 n}}{6 n} - \frac{3 a^{2} b x^{- 5 n}}{5 n} - \frac{3 a b^{2} x^{- 4 n}}{4 n} - \frac{b^{3} x^{- 3 n}}{3 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{3} \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 = 1.16838, size = 65, normalized size = 1.03 \begin{align*} -\frac{20 \, b^{3} x^{3 \, n} + 45 \, a b^{2} x^{2 \, n} + 36 \, a^{2} b x^{n} + 10 \, a^{3}}{60 \, n x^{6 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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