Optimal. Leaf size=44 \[ -\frac{a c}{x}+a d \log (x)+a e x+\frac{1}{2} b c x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4 \]
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Rubi [A] time = 0.0350813, 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} \[ -\frac{a c}{x}+a d \log (x)+a e x+\frac{1}{2} b c x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4 \]
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^2} \, dx &=\int \left (a e+\frac{a c}{x^2}+\frac{a d}{x}+b c x+b d x^2+b e x^3\right ) \, dx\\ &=-\frac{a c}{x}+a e x+\frac{1}{2} b c x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4+a d \log (x)\\ \end{align*}
Mathematica [A] time = 0.0048538, size = 44, normalized size = 1. \[ -\frac{a c}{x}+a d \log (x)+a e x+\frac{1}{2} b c x^2+\frac{1}{3} b d x^3+\frac{1}{4} b e x^4 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 39, normalized size = 0.9 \begin{align*} -{\frac{ac}{x}}+aex+{\frac{bc{x}^{2}}{2}}+{\frac{bd{x}^{3}}{3}}+{\frac{be{x}^{4}}{4}}+ad\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943359, size = 51, normalized size = 1.16 \begin{align*} \frac{1}{4} \, b e x^{4} + \frac{1}{3} \, b d x^{3} + \frac{1}{2} \, b c x^{2} + a e x + a d \log \left (x\right ) - \frac{a c}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43856, size = 113, normalized size = 2.57 \begin{align*} \frac{3 \, b e x^{5} + 4 \, b d x^{4} + 6 \, b c x^{3} + 12 \, a e x^{2} + 12 \, a d x \log \left (x\right ) - 12 \, a c}{12 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.296316, size = 41, normalized size = 0.93 \begin{align*} - \frac{a c}{x} + a d \log{\left (x \right )} + a e x + \frac{b c x^{2}}{2} + \frac{b d x^{3}}{3} + \frac{b e x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06735, size = 55, normalized size = 1.25 \begin{align*} \frac{1}{4} \, b x^{4} e + \frac{1}{3} \, b d x^{3} + \frac{1}{2} \, b c x^{2} + a x e + a d \log \left ({\left | x \right |}\right ) - \frac{a c}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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