Optimal. Leaf size=63 \[ -\frac{a^2 b x^{-6 n}}{2 n}-\frac{a^3 x^{-7 n}}{7 n}-\frac{3 a b^2 x^{-5 n}}{5 n}-\frac{b^3 x^{-4 n}}{4 n} \]
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Rubi [A] time = 0.0235806, 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} \[ -\frac{a^2 b x^{-6 n}}{2 n}-\frac{a^3 x^{-7 n}}{7 n}-\frac{3 a b^2 x^{-5 n}}{5 n}-\frac{b^3 x^{-4 n}}{4 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-7 n} \left (a+b x^n\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^8} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^3}{x^8}+\frac{3 a^2 b}{x^7}+\frac{3 a b^2}{x^6}+\frac{b^3}{x^5}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^3 x^{-7 n}}{7 n}-\frac{a^2 b x^{-6 n}}{2 n}-\frac{3 a b^2 x^{-5 n}}{5 n}-\frac{b^3 x^{-4 n}}{4 n}\\ \end{align*}
Mathematica [A] time = 0.0204867, size = 48, normalized size = 0.76 \[ -\frac{x^{-7 n} \left (70 a^2 b x^n+20 a^3+84 a b^2 x^{2 n}+35 b^3 x^{3 n}\right )}{140 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 56, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{4\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{3\,{b}^{2}a}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{b{a}^{2}}{2\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{{a}^{3}}{7\,n \left ({x}^{n} \right ) ^{7}}} \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 = 1.32103, size = 108, normalized size = 1.71 \begin{align*} -\frac{35 \, b^{3} x^{3 \, n} + 84 \, a b^{2} x^{2 \, n} + 70 \, a^{2} b x^{n} + 20 \, a^{3}}{140 \, n x^{7 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 31.7118, size = 61, normalized size = 0.97 \begin{align*} \begin{cases} - \frac{a^{3} x^{- 7 n}}{7 n} - \frac{a^{2} b x^{- 6 n}}{2 n} - \frac{3 a b^{2} x^{- 5 n}}{5 n} - \frac{b^{3} x^{- 4 n}}{4 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 = 1.19891, size = 65, normalized size = 1.03 \begin{align*} -\frac{35 \, b^{3} x^{3 \, n} + 84 \, a b^{2} x^{2 \, n} + 70 \, a^{2} b x^{n} + 20 \, a^{3}}{140 \, n x^{7 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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