Optimal. Leaf size=54 \[ \frac{a^2 x^5}{5}+\frac{1}{9} x^9 \left (2 a c+b^2\right )+\frac{2}{7} a b x^7+\frac{2}{11} b c x^{11}+\frac{c^2 x^{13}}{13} \]
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Rubi [A] time = 0.0360609, 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, 1108} \[ \frac{a^2 x^5}{5}+\frac{1}{9} x^9 \left (2 a c+b^2\right )+\frac{2}{7} a b x^7+\frac{2}{11} b c x^{11}+\frac{c^2 x^{13}}{13} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 1108
Rubi steps
\begin{align*} \int x^2 \left (a x+b x^3+c x^5\right )^2 \, dx &=\int x^4 \left (a+b x^2+c x^4\right )^2 \, dx\\ &=\int \left (a^2 x^4+2 a b x^6+\left (b^2+2 a c\right ) x^8+2 b c x^{10}+c^2 x^{12}\right ) \, dx\\ &=\frac{a^2 x^5}{5}+\frac{2}{7} a b x^7+\frac{1}{9} \left (b^2+2 a c\right ) x^9+\frac{2}{11} b c x^{11}+\frac{c^2 x^{13}}{13}\\ \end{align*}
Mathematica [A] time = 0.0070494, size = 54, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{1}{9} x^9 \left (2 a c+b^2\right )+\frac{2}{7} a b x^7+\frac{2}{11} b c x^{11}+\frac{c^2 x^{13}}{13} \]
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}^{5}}{5}}+{\frac{2\,ab{x}^{7}}{7}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{9}}{9}}+{\frac{2\,bc{x}^{11}}{11}}+{\frac{{c}^{2}{x}^{13}}{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11908, size = 59, normalized size = 1.09 \begin{align*} \frac{1}{13} \, c^{2} x^{13} + \frac{2}{11} \, b c x^{11} + \frac{1}{9} \,{\left (b^{2} + 2 \, a c\right )} x^{9} + \frac{2}{7} \, a b x^{7} + \frac{1}{5} \, a^{2} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.01494, size = 117, normalized size = 2.17 \begin{align*} \frac{1}{13} x^{13} c^{2} + \frac{2}{11} x^{11} c b + \frac{1}{9} x^{9} b^{2} + \frac{2}{9} x^{9} c a + \frac{2}{7} x^{7} b a + \frac{1}{5} x^{5} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.069509, size = 51, normalized size = 0.94 \begin{align*} \frac{a^{2} x^{5}}{5} + \frac{2 a b x^{7}}{7} + \frac{2 b c x^{11}}{11} + \frac{c^{2} x^{13}}{13} + x^{9} \left (\frac{2 a c}{9} + \frac{b^{2}}{9}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08607, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{13} \, c^{2} x^{13} + \frac{2}{11} \, b c x^{11} + \frac{1}{9} \, b^{2} x^{9} + \frac{2}{9} \, a c x^{9} + \frac{2}{7} \, a b x^{7} + \frac{1}{5} \, a^{2} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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