Optimal. Leaf size=54 \[ \frac{a^2 x^5}{5}+\frac{1}{7} x^7 \left (2 a c+b^2\right )+\frac{1}{3} a b x^6+\frac{1}{4} b c x^8+\frac{c^2 x^9}{9} \]
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Rubi [A] time = 0.0265261, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1594, 698} \[ \frac{a^2 x^5}{5}+\frac{1}{7} x^7 \left (2 a c+b^2\right )+\frac{1}{3} a b x^6+\frac{1}{4} b c x^8+\frac{c^2 x^9}{9} \]
Antiderivative was successfully verified.
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Rule 1594
Rule 698
Rubi steps
\begin{align*} \int \left (a x^2+b x^3+c x^4\right )^2 \, dx &=\int x^4 \left (a+b x+c x^2\right )^2 \, dx\\ &=\int \left (a^2 x^4+2 a b x^5+\left (b^2+2 a c\right ) x^6+2 b c x^7+c^2 x^8\right ) \, dx\\ &=\frac{a^2 x^5}{5}+\frac{1}{3} a b x^6+\frac{1}{7} \left (b^2+2 a c\right ) x^7+\frac{1}{4} b c x^8+\frac{c^2 x^9}{9}\\ \end{align*}
Mathematica [A] time = 0.0060407, size = 54, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{1}{7} x^7 \left (2 a c+b^2\right )+\frac{1}{3} a b x^6+\frac{1}{4} b c x^8+\frac{c^2 x^9}{9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{5}}{5}}+{\frac{ab{x}^{6}}{3}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{7}}{7}}+{\frac{bc{x}^{8}}{4}}+{\frac{{c}^{2}{x}^{9}}{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.33673, size = 65, normalized size = 1.2 \begin{align*} \frac{1}{9} \, c^{2} x^{9} + \frac{1}{4} \, b c x^{8} + \frac{1}{7} \, b^{2} x^{7} + \frac{1}{5} \, a^{2} x^{5} + \frac{1}{21} \,{\left (6 \, c x^{7} + 7 \, b x^{6}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36451, size = 112, normalized size = 2.07 \begin{align*} \frac{1}{9} x^{9} c^{2} + \frac{1}{4} x^{8} c b + \frac{1}{7} x^{7} b^{2} + \frac{2}{7} x^{7} c a + \frac{1}{3} x^{6} 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 = 0.073278, size = 48, normalized size = 0.89 \begin{align*} \frac{a^{2} x^{5}}{5} + \frac{a b x^{6}}{3} + \frac{b c x^{8}}{4} + \frac{c^{2} x^{9}}{9} + x^{7} \left (\frac{2 a c}{7} + \frac{b^{2}}{7}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09585, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{9} \, c^{2} x^{9} + \frac{1}{4} \, b c x^{8} + \frac{1}{7} \, b^{2} x^{7} + \frac{2}{7} \, a c x^{7} + \frac{1}{3} \, a b x^{6} + \frac{1}{5} \, a^{2} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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