Optimal. Leaf size=37 \[ -\frac{3 a^2 b}{x}+a^3 \log (x)-\frac{3 a b^2}{2 x^2}-\frac{b^3}{3 x^3} \]
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Rubi [A] time = 0.014899, antiderivative size = 37, 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} \[ -\frac{3 a^2 b}{x}+a^3 \log (x)-\frac{3 a b^2}{2 x^2}-\frac{b^3}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^3}{x} \, dx &=\int \frac{(b+a x)^3}{x^4} \, dx\\ &=\int \left (\frac{b^3}{x^4}+\frac{3 a b^2}{x^3}+\frac{3 a^2 b}{x^2}+\frac{a^3}{x}\right ) \, dx\\ &=-\frac{b^3}{3 x^3}-\frac{3 a b^2}{2 x^2}-\frac{3 a^2 b}{x}+a^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0035974, size = 37, normalized size = 1. \[ -\frac{3 a^2 b}{x}+a^3 \log (x)-\frac{3 a b^2}{2 x^2}-\frac{b^3}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{3\,{x}^{3}}}-{\frac{3\,{b}^{2}a}{2\,{x}^{2}}}-3\,{\frac{{a}^{2}b}{x}}+{a}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963716, size = 46, normalized size = 1.24 \begin{align*} a^{3} \log \left (x\right ) - \frac{18 \, a^{2} b x^{2} + 9 \, a b^{2} x + 2 \, b^{3}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45002, size = 85, normalized size = 2.3 \begin{align*} \frac{6 \, a^{3} x^{3} \log \left (x\right ) - 18 \, a^{2} b x^{2} - 9 \, a b^{2} x - 2 \, b^{3}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.336971, size = 34, normalized size = 0.92 \begin{align*} a^{3} \log{\left (x \right )} - \frac{18 a^{2} b x^{2} + 9 a b^{2} x + 2 b^{3}}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.58979, size = 47, normalized size = 1.27 \begin{align*} a^{3} \log \left ({\left | x \right |}\right ) - \frac{18 \, a^{2} b x^{2} + 9 \, a b^{2} x + 2 \, b^{3}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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