Optimal. Leaf size=30 \[ -\frac{a^2}{2 x^2}-\frac{2 a b}{3 x^3}-\frac{b^2}{4 x^4} \]
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Rubi [A] time = 0.01058, antiderivative size = 30, 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} \[ -\frac{a^2}{2 x^2}-\frac{2 a b}{3 x^3}-\frac{b^2}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^2}{x^3} \, dx &=\int \frac{(b+a x)^2}{x^5} \, dx\\ &=\int \left (\frac{b^2}{x^5}+\frac{2 a b}{x^4}+\frac{a^2}{x^3}\right ) \, dx\\ &=-\frac{b^2}{4 x^4}-\frac{2 a b}{3 x^3}-\frac{a^2}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0030134, size = 30, normalized size = 1. \[ -\frac{a^2}{2 x^2}-\frac{2 a b}{3 x^3}-\frac{b^2}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{b}^{2}}{4\,{x}^{4}}}-{\frac{2\,ab}{3\,{x}^{3}}}-{\frac{{a}^{2}}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96483, size = 32, normalized size = 1.07 \begin{align*} -\frac{6 \, a^{2} x^{2} + 8 \, a b x + 3 \, b^{2}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41532, size = 55, normalized size = 1.83 \begin{align*} -\frac{6 \, a^{2} x^{2} + 8 \, a b x + 3 \, b^{2}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.302161, size = 26, normalized size = 0.87 \begin{align*} - \frac{6 a^{2} x^{2} + 8 a b x + 3 b^{2}}{12 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18694, size = 32, normalized size = 1.07 \begin{align*} -\frac{6 \, a^{2} x^{2} + 8 \, a b x + 3 \, b^{2}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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