Optimal. Leaf size=46 \[ \frac{b^3}{a^4 (a x+b)}+\frac{3 b^2 \log (a x+b)}{a^4}-\frac{2 b x}{a^3}+\frac{x^2}{2 a^2} \]
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Rubi [A] time = 0.0281744, antiderivative size = 46, 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} \[ \frac{b^3}{a^4 (a x+b)}+\frac{3 b^2 \log (a x+b)}{a^4}-\frac{2 b x}{a^3}+\frac{x^2}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{\left (a+\frac{b}{x}\right )^2} \, dx &=\int \frac{x^3}{(b+a x)^2} \, dx\\ &=\int \left (-\frac{2 b}{a^3}+\frac{x}{a^2}-\frac{b^3}{a^3 (b+a x)^2}+\frac{3 b^2}{a^3 (b+a x)}\right ) \, dx\\ &=-\frac{2 b x}{a^3}+\frac{x^2}{2 a^2}+\frac{b^3}{a^4 (b+a x)}+\frac{3 b^2 \log (b+a x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0128846, size = 43, normalized size = 0.93 \[ \frac{a^2 x^2+\frac{2 b^3}{a x+b}+6 b^2 \log (a x+b)-4 a b x}{2 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 45, normalized size = 1. \begin{align*} -2\,{\frac{bx}{{a}^{3}}}+{\frac{{x}^{2}}{2\,{a}^{2}}}+{\frac{{b}^{3}}{{a}^{4} \left ( ax+b \right ) }}+3\,{\frac{{b}^{2}\ln \left ( ax+b \right ) }{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973541, size = 63, normalized size = 1.37 \begin{align*} \frac{b^{3}}{a^{5} x + a^{4} b} + \frac{3 \, b^{2} \log \left (a x + b\right )}{a^{4}} + \frac{a x^{2} - 4 \, b x}{2 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47135, size = 132, normalized size = 2.87 \begin{align*} \frac{a^{3} x^{3} - 3 \, a^{2} b x^{2} - 4 \, a b^{2} x + 2 \, b^{3} + 6 \,{\left (a b^{2} x + b^{3}\right )} \log \left (a x + b\right )}{2 \,{\left (a^{5} x + a^{4} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.323778, size = 44, normalized size = 0.96 \begin{align*} \frac{b^{3}}{a^{5} x + a^{4} b} + \frac{x^{2}}{2 a^{2}} - \frac{2 b x}{a^{3}} + \frac{3 b^{2} \log{\left (a x + b \right )}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09019, size = 65, normalized size = 1.41 \begin{align*} \frac{3 \, b^{2} \log \left ({\left | a x + b \right |}\right )}{a^{4}} + \frac{b^{3}}{{\left (a x + b\right )} a^{4}} + \frac{a^{2} x^{2} - 4 \, a b x}{2 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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