Optimal. Leaf size=74 \[ \frac{1}{4} x^4 (a C+A b)+\frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{6} x^6 (A c+b C)+\frac{1}{5} b B x^5+\frac{1}{7} B c x^7+\frac{1}{8} c C x^8 \]
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Rubi [A] time = 0.0565914, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1628} \[ \frac{1}{4} x^4 (a C+A b)+\frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{6} x^6 (A c+b C)+\frac{1}{5} b B x^5+\frac{1}{7} B c x^7+\frac{1}{8} c C x^8 \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin{align*} \int x \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a A x+a B x^2+(A b+a C) x^3+b B x^4+(A c+b C) x^5+B c x^6+c C x^7\right ) \, dx\\ &=\frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{4} (A b+a C) x^4+\frac{1}{5} b B x^5+\frac{1}{6} (A c+b C) x^6+\frac{1}{7} B c x^7+\frac{1}{8} c C x^8\\ \end{align*}
Mathematica [A] time = 0.0120258, size = 74, normalized size = 1. \[ \frac{1}{4} x^4 (a C+A b)+\frac{1}{2} a A x^2+\frac{1}{3} a B x^3+\frac{1}{6} x^6 (A c+b C)+\frac{1}{5} b B x^5+\frac{1}{7} B c x^7+\frac{1}{8} c C x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 61, normalized size = 0.8 \begin{align*}{\frac{aA{x}^{2}}{2}}+{\frac{aB{x}^{3}}{3}}+{\frac{ \left ( Ab+aC \right ){x}^{4}}{4}}+{\frac{bB{x}^{5}}{5}}+{\frac{ \left ( Ac+bC \right ){x}^{6}}{6}}+{\frac{Bc{x}^{7}}{7}}+{\frac{cC{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961008, size = 81, normalized size = 1.09 \begin{align*} \frac{1}{8} \, C c x^{8} + \frac{1}{7} \, B c x^{7} + \frac{1}{5} \, B b x^{5} + \frac{1}{6} \,{\left (C b + A c\right )} x^{6} + \frac{1}{3} \, B a x^{3} + \frac{1}{4} \,{\left (C a + A b\right )} x^{4} + \frac{1}{2} \, A a x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.12357, size = 169, normalized size = 2.28 \begin{align*} \frac{1}{8} x^{8} c C + \frac{1}{7} x^{7} c B + \frac{1}{6} x^{6} b C + \frac{1}{6} x^{6} c A + \frac{1}{5} x^{5} b B + \frac{1}{4} x^{4} a C + \frac{1}{4} x^{4} b A + \frac{1}{3} x^{3} a B + \frac{1}{2} x^{2} a A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.07363, size = 68, normalized size = 0.92 \begin{align*} \frac{A a x^{2}}{2} + \frac{B a x^{3}}{3} + \frac{B b x^{5}}{5} + \frac{B c x^{7}}{7} + \frac{C c x^{8}}{8} + x^{6} \left (\frac{A c}{6} + \frac{C b}{6}\right ) + x^{4} \left (\frac{A b}{4} + \frac{C a}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08453, size = 86, normalized size = 1.16 \begin{align*} \frac{1}{8} \, C c x^{8} + \frac{1}{7} \, B c x^{7} + \frac{1}{6} \, C b x^{6} + \frac{1}{6} \, A c x^{6} + \frac{1}{5} \, B b x^{5} + \frac{1}{4} \, C a x^{4} + \frac{1}{4} \, A b x^{4} + \frac{1}{3} \, B a x^{3} + \frac{1}{2} \, A a x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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