Optimal. Leaf size=44 \[ -\frac{a c}{2 x^2}-\frac{a d}{x}+a e \log (x)+b c x+\frac{1}{2} b d x^2+\frac{1}{3} b e x^3 \]
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Rubi [A] time = 0.0341978, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {1628} \[ -\frac{a c}{2 x^2}-\frac{a d}{x}+a e \log (x)+b c x+\frac{1}{2} b d x^2+\frac{1}{3} b e x^3 \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin{align*} \int \frac{\left (c+d x+e x^2\right ) \left (a+b x^3\right )}{x^3} \, dx &=\int \left (b c+\frac{a c}{x^3}+\frac{a d}{x^2}+\frac{a e}{x}+b d x+b e x^2\right ) \, dx\\ &=-\frac{a c}{2 x^2}-\frac{a d}{x}+b c x+\frac{1}{2} b d x^2+\frac{1}{3} b e x^3+a e \log (x)\\ \end{align*}
Mathematica [A] time = 0.0040648, size = 44, normalized size = 1. \[ -\frac{a c}{2 x^2}-\frac{a d}{x}+a e \log (x)+b c x+\frac{1}{2} b d x^2+\frac{1}{3} b e x^3 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 39, normalized size = 0.9 \begin{align*} -{\frac{ac}{2\,{x}^{2}}}-{\frac{ad}{x}}+bcx+{\frac{bd{x}^{2}}{2}}+{\frac{be{x}^{3}}{3}}+ae\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.937409, size = 51, normalized size = 1.16 \begin{align*} \frac{1}{3} \, b e x^{3} + \frac{1}{2} \, b d x^{2} + b c x + a e \log \left (x\right ) - \frac{2 \, a d x + a c}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50428, size = 111, normalized size = 2.52 \begin{align*} \frac{2 \, b e x^{5} + 3 \, b d x^{4} + 6 \, b c x^{3} + 6 \, a e x^{2} \log \left (x\right ) - 6 \, a d x - 3 \, a c}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.365596, size = 42, normalized size = 0.95 \begin{align*} a e \log{\left (x \right )} + b c x + \frac{b d x^{2}}{2} + \frac{b e x^{3}}{3} - \frac{a c + 2 a d x}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06201, size = 55, normalized size = 1.25 \begin{align*} \frac{1}{3} \, b x^{3} e + \frac{1}{2} \, b d x^{2} + b c x + a e \log \left ({\left | x \right |}\right ) - \frac{2 \, a d x + a c}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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