Optimal. Leaf size=43 \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{5 x^5}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{8 x^8} \]
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Rubi [A] time = 0.0145501, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 43} \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{5 x^5}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{8 x^8} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^3}{x^6} \, dx &=\int \frac{(b+a x)^3}{x^9} \, dx\\ &=\int \left (\frac{b^3}{x^9}+\frac{3 a b^2}{x^8}+\frac{3 a^2 b}{x^7}+\frac{a^3}{x^6}\right ) \, dx\\ &=-\frac{b^3}{8 x^8}-\frac{3 a b^2}{7 x^7}-\frac{a^2 b}{2 x^6}-\frac{a^3}{5 x^5}\\ \end{align*}
Mathematica [A] time = 0.0059075, size = 43, normalized size = 1. \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{5 x^5}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{8 x^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}}{8\,{x}^{8}}}-{\frac{3\,{b}^{2}a}{7\,{x}^{7}}}-{\frac{{a}^{2}b}{2\,{x}^{6}}}-{\frac{{a}^{3}}{5\,{x}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985202, size = 47, normalized size = 1.09 \begin{align*} -\frac{56 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 120 \, a b^{2} x + 35 \, b^{3}}{280 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39167, size = 86, normalized size = 2. \begin{align*} -\frac{56 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 120 \, a b^{2} x + 35 \, b^{3}}{280 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.413283, size = 37, normalized size = 0.86 \begin{align*} - \frac{56 a^{3} x^{3} + 140 a^{2} b x^{2} + 120 a b^{2} x + 35 b^{3}}{280 x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20866, size = 47, normalized size = 1.09 \begin{align*} -\frac{56 \, a^{3} x^{3} + 140 \, a^{2} b x^{2} + 120 \, a b^{2} x + 35 \, b^{3}}{280 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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