Optimal. Leaf size=35 \[ \frac{3}{2} a^2 b x^2+\frac{a^3 x^3}{3}+3 a b^2 x+b^3 \log (x) \]
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Rubi [A] time = 0.011661, antiderivative size = 35, 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} \[ \frac{3}{2} a^2 b x^2+\frac{a^3 x^3}{3}+3 a b^2 x+b^3 \log (x) \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^3 x^2 \, dx &=\int \frac{(b+a x)^3}{x} \, dx\\ &=\int \left (3 a b^2+\frac{b^3}{x}+3 a^2 b x+a^3 x^2\right ) \, dx\\ &=3 a b^2 x+\frac{3}{2} a^2 b x^2+\frac{a^3 x^3}{3}+b^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0027684, size = 35, normalized size = 1. \[ \frac{3}{2} a^2 b x^2+\frac{a^3 x^3}{3}+3 a b^2 x+b^3 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 32, normalized size = 0.9 \begin{align*} 3\,xa{b}^{2}+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{\frac{{a}^{3}{x}^{3}}{3}}+{b}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997642, size = 42, normalized size = 1.2 \begin{align*} \frac{1}{3} \, a^{3} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4567, size = 73, normalized size = 2.09 \begin{align*} \frac{1}{3} \, a^{3} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.257519, size = 34, normalized size = 0.97 \begin{align*} \frac{a^{3} x^{3}}{3} + \frac{3 a^{2} b x^{2}}{2} + 3 a b^{2} x + b^{3} \log{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17491, size = 43, normalized size = 1.23 \begin{align*} \frac{1}{3} \, a^{3} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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