Optimal. Leaf size=24 \[ a^2 \log (x)-\frac{2 a b}{x}-\frac{b^2}{2 x^2} \]
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Rubi [A] time = 0.0105294, antiderivative size = 24, 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} \[ a^2 \log (x)-\frac{2 a b}{x}-\frac{b^2}{2 x^2} \]
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} \, dx &=\int \frac{(b+a x)^2}{x^3} \, dx\\ &=\int \left (\frac{b^2}{x^3}+\frac{2 a b}{x^2}+\frac{a^2}{x}\right ) \, dx\\ &=-\frac{b^2}{2 x^2}-\frac{2 a b}{x}+a^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0034823, size = 24, normalized size = 1. \[ a^2 \log (x)-\frac{2 a b}{x}-\frac{b^2}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 23, normalized size = 1. \begin{align*} -{\frac{{b}^{2}}{2\,{x}^{2}}}-2\,{\frac{ab}{x}}+{a}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952182, size = 28, normalized size = 1.17 \begin{align*} a^{2} \log \left (x\right ) - \frac{4 \, a b x + b^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44283, size = 59, normalized size = 2.46 \begin{align*} \frac{2 \, a^{2} x^{2} \log \left (x\right ) - 4 \, a b x - b^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.291155, size = 20, normalized size = 0.83 \begin{align*} a^{2} \log{\left (x \right )} - \frac{4 a b x + b^{2}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15572, size = 30, normalized size = 1.25 \begin{align*} a^{2} \log \left ({\left | x \right |}\right ) - \frac{4 \, a b x + b^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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