Optimal. Leaf size=22 \[ \frac{a^2 x^2}{2}+2 a b x+b^2 \log (x) \]
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Rubi [A] time = 0.0080695, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {263, 43} \[ \frac{a^2 x^2}{2}+2 a b x+b^2 \log (x) \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^2 x \, dx &=\int \frac{(b+a x)^2}{x} \, dx\\ &=\int \left (2 a b+\frac{b^2}{x}+a^2 x\right ) \, dx\\ &=2 a b x+\frac{a^2 x^2}{2}+b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.000959, size = 22, normalized size = 1. \[ \frac{a^2 x^2}{2}+2 a b x+b^2 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 21, normalized size = 1. \begin{align*} 2\,xab+{\frac{{a}^{2}{x}^{2}}{2}}+{b}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948466, size = 27, normalized size = 1.23 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + 2 \, a b x + b^{2} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46167, size = 49, normalized size = 2.23 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + 2 \, a b x + b^{2} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.246043, size = 20, normalized size = 0.91 \begin{align*} \frac{a^{2} x^{2}}{2} + 2 a b x + b^{2} \log{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13985, size = 28, normalized size = 1.27 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + 2 \, a b x + b^{2} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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