Optimal. Leaf size=43 \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{4 x^4}-\frac{a b^2}{2 x^6}-\frac{b^3}{7 x^7} \]
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Rubi [A] time = 0.0149061, 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} \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{4 x^4}-\frac{a b^2}{2 x^6}-\frac{b^3}{7 x^7} \]
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^5} \, dx &=\int \frac{(b+a x)^3}{x^8} \, dx\\ &=\int \left (\frac{b^3}{x^8}+\frac{3 a b^2}{x^7}+\frac{3 a^2 b}{x^6}+\frac{a^3}{x^5}\right ) \, dx\\ &=-\frac{b^3}{7 x^7}-\frac{a b^2}{2 x^6}-\frac{3 a^2 b}{5 x^5}-\frac{a^3}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.0032526, size = 43, normalized size = 1. \[ -\frac{3 a^2 b}{5 x^5}-\frac{a^3}{4 x^4}-\frac{a b^2}{2 x^6}-\frac{b^3}{7 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}}{7\,{x}^{7}}}-{\frac{{b}^{2}a}{2\,{x}^{6}}}-{\frac{3\,{a}^{2}b}{5\,{x}^{5}}}-{\frac{{a}^{3}}{4\,{x}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.013, size = 47, normalized size = 1.09 \begin{align*} -\frac{35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4386, size = 84, normalized size = 1.95 \begin{align*} -\frac{35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.393709, size = 37, normalized size = 0.86 \begin{align*} - \frac{35 a^{3} x^{3} + 84 a^{2} b x^{2} + 70 a b^{2} x + 20 b^{3}}{140 x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21555, size = 47, normalized size = 1.09 \begin{align*} -\frac{35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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