Optimal. Leaf size=54 \[ \frac{a^2 x^6}{6}+\frac{1}{8} x^8 \left (2 a c+b^2\right )+\frac{2}{7} a b x^7+\frac{2}{9} b c x^9+\frac{c^2 x^{10}}{10} \]
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Rubi [A] time = 0.028877, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1585, 698} \[ \frac{a^2 x^6}{6}+\frac{1}{8} x^8 \left (2 a c+b^2\right )+\frac{2}{7} a b x^7+\frac{2}{9} b c x^9+\frac{c^2 x^{10}}{10} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 698
Rubi steps
\begin{align*} \int x \left (a x^2+b x^3+c x^4\right )^2 \, dx &=\int x^5 \left (a+b x+c x^2\right )^2 \, dx\\ &=\int \left (a^2 x^5+2 a b x^6+\left (b^2+2 a c\right ) x^7+2 b c x^8+c^2 x^9\right ) \, dx\\ &=\frac{a^2 x^6}{6}+\frac{2}{7} a b x^7+\frac{1}{8} \left (b^2+2 a c\right ) x^8+\frac{2}{9} b c x^9+\frac{c^2 x^{10}}{10}\\ \end{align*}
Mathematica [A] time = 0.0068741, size = 54, normalized size = 1. \[ \frac{a^2 x^6}{6}+\frac{1}{8} x^8 \left (2 a c+b^2\right )+\frac{2}{7} a b x^7+\frac{2}{9} b c x^9+\frac{c^2 x^{10}}{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{6}}{6}}+{\frac{2\,ab{x}^{7}}{7}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{8}}{8}}+{\frac{2\,bc{x}^{9}}{9}}+{\frac{{c}^{2}{x}^{10}}{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12733, size = 59, normalized size = 1.09 \begin{align*} \frac{1}{10} \, c^{2} x^{10} + \frac{2}{9} \, b c x^{9} + \frac{2}{7} \, a b x^{7} + \frac{1}{8} \,{\left (b^{2} + 2 \, a c\right )} x^{8} + \frac{1}{6} \, a^{2} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28945, size = 115, normalized size = 2.13 \begin{align*} \frac{1}{10} x^{10} c^{2} + \frac{2}{9} x^{9} c b + \frac{1}{8} x^{8} b^{2} + \frac{1}{4} x^{8} c a + \frac{2}{7} x^{7} b a + \frac{1}{6} x^{6} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.069998, size = 49, normalized size = 0.91 \begin{align*} \frac{a^{2} x^{6}}{6} + \frac{2 a b x^{7}}{7} + \frac{2 b c x^{9}}{9} + \frac{c^{2} x^{10}}{10} + x^{8} \left (\frac{a c}{4} + \frac{b^{2}}{8}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08124, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{10} \, c^{2} x^{10} + \frac{2}{9} \, b c x^{9} + \frac{1}{8} \, b^{2} x^{8} + \frac{1}{4} \, a c x^{8} + \frac{2}{7} \, a b x^{7} + \frac{1}{6} \, a^{2} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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