Optimal. Leaf size=77 \[ \frac{b^5}{2 a^6 (a x+b)^2}-\frac{5 b^4}{a^6 (a x+b)}+\frac{6 b^2 x}{a^5}-\frac{10 b^3 \log (a x+b)}{a^6}-\frac{3 b x^2}{2 a^4}+\frac{x^3}{3 a^3} \]
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Rubi [A] time = 0.0491546, antiderivative size = 77, 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} \[ \frac{b^5}{2 a^6 (a x+b)^2}-\frac{5 b^4}{a^6 (a x+b)}+\frac{6 b^2 x}{a^5}-\frac{10 b^3 \log (a x+b)}{a^6}-\frac{3 b x^2}{2 a^4}+\frac{x^3}{3 a^3} \]
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 )^3} \, dx &=\int \frac{x^5}{(b+a x)^3} \, dx\\ &=\int \left (\frac{6 b^2}{a^5}-\frac{3 b x}{a^4}+\frac{x^2}{a^3}-\frac{b^5}{a^5 (b+a x)^3}+\frac{5 b^4}{a^5 (b+a x)^2}-\frac{10 b^3}{a^5 (b+a x)}\right ) \, dx\\ &=\frac{6 b^2 x}{a^5}-\frac{3 b x^2}{2 a^4}+\frac{x^3}{3 a^3}+\frac{b^5}{2 a^6 (b+a x)^2}-\frac{5 b^4}{a^6 (b+a x)}-\frac{10 b^3 \log (b+a x)}{a^6}\\ \end{align*}
Mathematica [A] time = 0.0414243, size = 63, normalized size = 0.82 \[ \frac{-9 a^2 b x^2+2 a^3 x^3-\frac{3 b^4 (10 a x+9 b)}{(a x+b)^2}+36 a b^2 x-60 b^3 \log (a x+b)}{6 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 72, normalized size = 0.9 \begin{align*} 6\,{\frac{{b}^{2}x}{{a}^{5}}}-{\frac{3\,b{x}^{2}}{2\,{a}^{4}}}+{\frac{{x}^{3}}{3\,{a}^{3}}}+{\frac{{b}^{5}}{2\,{a}^{6} \left ( ax+b \right ) ^{2}}}-5\,{\frac{{b}^{4}}{{a}^{6} \left ( ax+b \right ) }}-10\,{\frac{{b}^{3}\ln \left ( ax+b \right ) }{{a}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01018, size = 109, normalized size = 1.42 \begin{align*} -\frac{10 \, a b^{4} x + 9 \, b^{5}}{2 \,{\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}} - \frac{10 \, b^{3} \log \left (a x + b\right )}{a^{6}} + \frac{2 \, a^{2} x^{3} - 9 \, a b x^{2} + 36 \, b^{2} x}{6 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43982, size = 227, normalized size = 2.95 \begin{align*} \frac{2 \, a^{5} x^{5} - 5 \, a^{4} b x^{4} + 20 \, a^{3} b^{2} x^{3} + 63 \, a^{2} b^{3} x^{2} + 6 \, a b^{4} x - 27 \, b^{5} - 60 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )} \log \left (a x + b\right )}{6 \,{\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.441964, size = 83, normalized size = 1.08 \begin{align*} - \frac{10 a b^{4} x + 9 b^{5}}{2 a^{8} x^{2} + 4 a^{7} b x + 2 a^{6} b^{2}} + \frac{x^{3}}{3 a^{3}} - \frac{3 b x^{2}}{2 a^{4}} + \frac{6 b^{2} x}{a^{5}} - \frac{10 b^{3} \log{\left (a x + b \right )}}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08795, size = 99, normalized size = 1.29 \begin{align*} -\frac{10 \, b^{3} \log \left ({\left | a x + b \right |}\right )}{a^{6}} - \frac{10 \, a b^{4} x + 9 \, b^{5}}{2 \,{\left (a x + b\right )}^{2} a^{6}} + \frac{2 \, a^{6} x^{3} - 9 \, a^{5} b x^{2} + 36 \, a^{4} b^{2} x}{6 \, a^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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