Optimal. Leaf size=58 \[ -\frac{b^4}{a^5 (a x+b)}+\frac{3 b^2 x}{a^4}-\frac{4 b^3 \log (a x+b)}{a^5}-\frac{b x^2}{a^3}+\frac{x^3}{3 a^2} \]
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Rubi [A] time = 0.0370357, antiderivative size = 58, 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} \[ -\frac{b^4}{a^5 (a x+b)}+\frac{3 b^2 x}{a^4}-\frac{4 b^3 \log (a x+b)}{a^5}-\frac{b x^2}{a^3}+\frac{x^3}{3 a^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+\frac{b}{x}\right )^2} \, dx &=\int \frac{x^4}{(b+a x)^2} \, dx\\ &=\int \left (\frac{3 b^2}{a^4}-\frac{2 b x}{a^3}+\frac{x^2}{a^2}+\frac{b^4}{a^4 (b+a x)^2}-\frac{4 b^3}{a^4 (b+a x)}\right ) \, dx\\ &=\frac{3 b^2 x}{a^4}-\frac{b x^2}{a^3}+\frac{x^3}{3 a^2}-\frac{b^4}{a^5 (b+a x)}-\frac{4 b^3 \log (b+a x)}{a^5}\\ \end{align*}
Mathematica [A] time = 0.0188467, size = 54, normalized size = 0.93 \[ \frac{-3 a^2 b x^2+a^3 x^3-\frac{3 b^4}{a x+b}+9 a b^2 x-12 b^3 \log (a x+b)}{3 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 57, normalized size = 1. \begin{align*} 3\,{\frac{{b}^{2}x}{{a}^{4}}}-{\frac{b{x}^{2}}{{a}^{3}}}+{\frac{{x}^{3}}{3\,{a}^{2}}}-{\frac{{b}^{4}}{{a}^{5} \left ( ax+b \right ) }}-4\,{\frac{{b}^{3}\ln \left ( ax+b \right ) }{{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989066, size = 80, normalized size = 1.38 \begin{align*} -\frac{b^{4}}{a^{6} x + a^{5} b} - \frac{4 \, b^{3} \log \left (a x + b\right )}{a^{5}} + \frac{a^{2} x^{3} - 3 \, a b x^{2} + 9 \, b^{2} x}{3 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40926, size = 155, normalized size = 2.67 \begin{align*} \frac{a^{4} x^{4} - 2 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a b^{3} x - 3 \, b^{4} - 12 \,{\left (a b^{3} x + b^{4}\right )} \log \left (a x + b\right )}{3 \,{\left (a^{6} x + a^{5} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.340239, size = 54, normalized size = 0.93 \begin{align*} - \frac{b^{4}}{a^{6} x + a^{5} b} + \frac{x^{3}}{3 a^{2}} - \frac{b x^{2}}{a^{3}} + \frac{3 b^{2} x}{a^{4}} - \frac{4 b^{3} \log{\left (a x + b \right )}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12542, size = 84, normalized size = 1.45 \begin{align*} -\frac{4 \, b^{3} \log \left ({\left | a x + b \right |}\right )}{a^{5}} - \frac{b^{4}}{{\left (a x + b\right )} a^{5}} + \frac{a^{4} x^{3} - 3 \, a^{3} b x^{2} + 9 \, a^{2} b^{2} x}{3 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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