Optimal. Leaf size=33 \[ -\frac{3 a^2 b}{x}-\frac{a^3}{2 x^2}+3 a b^2 \log (x)+b^3 x \]
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Rubi [A] time = 0.0126738, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ -\frac{3 a^2 b}{x}-\frac{a^3}{2 x^2}+3 a b^2 \log (x)+b^3 x \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^3}{x^3} \, dx &=\int \left (b^3+\frac{a^3}{x^3}+\frac{3 a^2 b}{x^2}+\frac{3 a b^2}{x}\right ) \, dx\\ &=-\frac{a^3}{2 x^2}-\frac{3 a^2 b}{x}+b^3 x+3 a b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0071219, size = 33, normalized size = 1. \[ -\frac{3 a^2 b}{x}-\frac{a^3}{2 x^2}+3 a b^2 \log (x)+b^3 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 32, normalized size = 1. \begin{align*} -{\frac{{a}^{3}}{2\,{x}^{2}}}-3\,{\frac{{a}^{2}b}{x}}+{b}^{3}x+3\,a{b}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0208, size = 41, normalized size = 1.24 \begin{align*} b^{3} x + 3 \, a b^{2} \log \left (x\right ) - \frac{6 \, a^{2} b x + a^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57627, size = 81, normalized size = 2.45 \begin{align*} \frac{2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} \log \left (x\right ) - 6 \, a^{2} b x - a^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.44382, size = 31, normalized size = 0.94 \begin{align*} 3 a b^{2} \log{\left (x \right )} + b^{3} x - \frac{a^{3} + 6 a^{2} b x}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14986, size = 42, normalized size = 1.27 \begin{align*} b^{3} x + 3 \, a b^{2} \log \left ({\left | x \right |}\right ) - \frac{6 \, a^{2} b x + a^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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