Optimal. Leaf size=54 \[ \frac{a^2 x^2}{2}+\frac{1}{6} x^6 \left (2 a c+b^2\right )+\frac{1}{2} a b x^4+\frac{1}{4} b c x^8+\frac{c^2 x^{10}}{10} \]
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Rubi [A] time = 0.0383747, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1107, 611} \[ \frac{a^2 x^2}{2}+\frac{1}{6} x^6 \left (2 a c+b^2\right )+\frac{1}{2} a b x^4+\frac{1}{4} b c x^8+\frac{c^2 x^{10}}{10} \]
Antiderivative was successfully verified.
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Rule 1107
Rule 611
Rubi steps
\begin{align*} \int x \left (a+b x^2+c x^4\right )^2 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a+b x+c x^2\right )^2 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^2+2 a b x+b^2 \left (1+\frac{2 a c}{b^2}\right ) x^2+2 b c x^3+c^2 x^4\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 x^2}{2}+\frac{1}{2} a b x^4+\frac{1}{6} \left (b^2+2 a c\right ) x^6+\frac{1}{4} b c x^8+\frac{c^2 x^{10}}{10}\\ \end{align*}
Mathematica [A] time = 0.0078386, size = 48, normalized size = 0.89 \[ \frac{1}{60} x^2 \left (30 a^2+10 x^4 \left (2 a c+b^2\right )+30 a b x^2+15 b c x^6+6 c^2 x^8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{ab{x}^{4}}{2}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{6}}{6}}+{\frac{bc{x}^{8}}{4}}+{\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 = 0.953357, size = 59, normalized size = 1.09 \begin{align*} \frac{1}{10} \, c^{2} x^{10} + \frac{1}{4} \, b c x^{8} + \frac{1}{6} \,{\left (b^{2} + 2 \, a c\right )} x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, a^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.279, size = 115, normalized size = 2.13 \begin{align*} \frac{1}{10} x^{10} c^{2} + \frac{1}{4} x^{8} c b + \frac{1}{6} x^{6} b^{2} + \frac{1}{3} x^{6} c a + \frac{1}{2} x^{4} b a + \frac{1}{2} x^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.077345, size = 46, normalized size = 0.85 \begin{align*} \frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b c x^{8}}{4} + \frac{c^{2} x^{10}}{10} + 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 = 1.35185, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{10} \, c^{2} x^{10} + \frac{1}{4} \, b c x^{8} + \frac{1}{6} \, b^{2} x^{6} + \frac{1}{3} \, a c x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{2} \, a^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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