Optimal. Leaf size=31 \[ \frac{b^2 \log (a x+b)}{a^3}-\frac{b x}{a^2}+\frac{x^2}{2 a} \]
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Rubi [A] time = 0.0157775, antiderivative size = 31, 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} \[ \frac{b^2 \log (a x+b)}{a^3}-\frac{b x}{a^2}+\frac{x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{a+\frac{b}{x}} \, dx &=\int \frac{x^2}{b+a x} \, dx\\ &=\int \left (-\frac{b}{a^2}+\frac{x}{a}+\frac{b^2}{a^2 (b+a x)}\right ) \, dx\\ &=-\frac{b x}{a^2}+\frac{x^2}{2 a}+\frac{b^2 \log (b+a x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0031411, size = 31, normalized size = 1. \[ \frac{b^2 \log (a x+b)}{a^3}-\frac{b x}{a^2}+\frac{x^2}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 30, normalized size = 1. \begin{align*} -{\frac{bx}{{a}^{2}}}+{\frac{{x}^{2}}{2\,a}}+{\frac{{b}^{2}\ln \left ( ax+b \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991435, size = 39, normalized size = 1.26 \begin{align*} \frac{b^{2} \log \left (a x + b\right )}{a^{3}} + \frac{a x^{2} - 2 \, b x}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44309, size = 68, normalized size = 2.19 \begin{align*} \frac{a^{2} x^{2} - 2 \, a b x + 2 \, b^{2} \log \left (a x + b\right )}{2 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.259465, size = 26, normalized size = 0.84 \begin{align*} \frac{x^{2}}{2 a} - \frac{b x}{a^{2}} + \frac{b^{2} \log{\left (a x + b \right )}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09857, size = 41, normalized size = 1.32 \begin{align*} \frac{b^{2} \log \left ({\left | a x + b \right |}\right )}{a^{3}} + \frac{a x^{2} - 2 \, b x}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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