Optimal. Leaf size=54 \[ \frac{a^2 x^4}{4}+\frac{1}{8} x^8 \left (2 a c+b^2\right )+\frac{1}{3} a b x^6+\frac{1}{5} b c x^{10}+\frac{c^2 x^{12}}{12} \]
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Rubi [A] time = 0.0544296, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1585, 1114, 631} \[ \frac{a^2 x^4}{4}+\frac{1}{8} x^8 \left (2 a c+b^2\right )+\frac{1}{3} a b x^6+\frac{1}{5} b c x^{10}+\frac{c^2 x^{12}}{12} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 1114
Rule 631
Rubi steps
\begin{align*} \int x \left (a x+b x^3+c x^5\right )^2 \, dx &=\int x^3 \left (a+b x^2+c x^4\right )^2 \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x \left (a+b x+c x^2\right )^2 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^2 x+2 a b x^2+\left (b^2+2 a c\right ) x^3+2 b c x^4+c^2 x^5\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{1}{8} \left (b^2+2 a c\right ) x^8+\frac{1}{5} b c x^{10}+\frac{c^2 x^{12}}{12}\\ \end{align*}
Mathematica [A] time = 0.0092113, size = 48, normalized size = 0.89 \[ \frac{1}{120} x^4 \left (30 a^2+15 x^4 \left (2 a c+b^2\right )+40 a b x^2+24 b c x^6+10 c^2 x^8\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{ab{x}^{6}}{3}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{8}}{8}}+{\frac{bc{x}^{10}}{5}}+{\frac{{c}^{2}{x}^{12}}{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0896, size = 59, normalized size = 1.09 \begin{align*} \frac{1}{12} \, c^{2} x^{12} + \frac{1}{5} \, b c x^{10} + \frac{1}{8} \,{\left (b^{2} + 2 \, a c\right )} x^{8} + \frac{1}{3} \, a b 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 = 1.1233, size = 116, normalized size = 2.15 \begin{align*} \frac{1}{12} x^{12} c^{2} + \frac{1}{5} x^{10} c b + \frac{1}{8} x^{8} b^{2} + \frac{1}{4} x^{8} c a + \frac{1}{3} x^{6} b a + \frac{1}{4} x^{4} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.069945, size = 46, normalized size = 0.85 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{a b x^{6}}{3} + \frac{b c x^{10}}{5} + \frac{c^{2} x^{12}}{12} + 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 = 1.08953, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{12} \, c^{2} x^{12} + \frac{1}{5} \, b c x^{10} + \frac{1}{8} \, b^{2} x^{8} + \frac{1}{4} \, a c x^{8} + \frac{1}{3} \, a b 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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