Optimal. Leaf size=54 \[ \frac{a^2 x^4}{4}+\frac{1}{6} x^6 \left (2 a c+b^2\right )+\frac{2}{5} a b x^5+\frac{2}{7} b c x^7+\frac{c^2 x^8}{8} \]
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Rubi [A] time = 0.0320555, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1585, 698} \[ \frac{a^2 x^4}{4}+\frac{1}{6} x^6 \left (2 a c+b^2\right )+\frac{2}{5} a b x^5+\frac{2}{7} b c x^7+\frac{c^2 x^8}{8} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 698
Rubi steps
\begin{align*} \int \frac{\left (a x^2+b x^3+c x^4\right )^2}{x} \, dx &=\int x^3 \left (a+b x+c x^2\right )^2 \, dx\\ &=\int \left (a^2 x^3+2 a b x^4+\left (b^2+2 a c\right ) x^5+2 b c x^6+c^2 x^7\right ) \, dx\\ &=\frac{a^2 x^4}{4}+\frac{2}{5} a b x^5+\frac{1}{6} \left (b^2+2 a c\right ) x^6+\frac{2}{7} b c x^7+\frac{c^2 x^8}{8}\\ \end{align*}
Mathematica [A] time = 0.0062127, size = 54, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{1}{6} x^6 \left (2 a c+b^2\right )+\frac{2}{5} a b x^5+\frac{2}{7} b c x^7+\frac{c^2 x^8}{8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{2\,ab{x}^{5}}{5}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{6}}{6}}+{\frac{2\,bc{x}^{7}}{7}}+{\frac{{c}^{2}{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12473, size = 59, normalized size = 1.09 \begin{align*} \frac{1}{8} \, c^{2} x^{8} + \frac{2}{7} \, b c x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{6} \,{\left (b^{2} + 2 \, a c\right )} x^{6} + \frac{1}{4} \, a^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42506, size = 107, normalized size = 1.98 \begin{align*} \frac{1}{8} \, c^{2} x^{8} + \frac{2}{7} \, b c x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{6} \,{\left (b^{2} + 2 \, a c\right )} x^{6} + \frac{1}{4} \, a^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.070307, size = 49, normalized size = 0.91 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{2 a b x^{5}}{5} + \frac{2 b c x^{7}}{7} + \frac{c^{2} x^{8}}{8} + x^{6} \left (\frac{a c}{3} + \frac{b^{2}}{6}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09907, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{8} \, c^{2} x^{8} + \frac{2}{7} \, b c x^{7} + \frac{1}{6} \, b^{2} x^{6} + \frac{1}{3} \, a c x^{6} + \frac{2}{5} \, a b x^{5} + \frac{1}{4} \, a^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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