Optimal. Leaf size=65 \[ \frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{2}{5} a A c x^5+\frac{1}{3} a B c x^6+\frac{1}{7} A c^2 x^7+\frac{1}{8} B c^2 x^8 \]
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Rubi [A] time = 0.0528254, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ \frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{2}{5} a A c x^5+\frac{1}{3} a B c x^6+\frac{1}{7} A c^2 x^7+\frac{1}{8} B c^2 x^8 \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int x^2 (A+B x) \left (a+c x^2\right )^2 \, dx &=\int \left (a^2 A x^2+a^2 B x^3+2 a A c x^4+2 a B c x^5+A c^2 x^6+B c^2 x^7\right ) \, dx\\ &=\frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{2}{5} a A c x^5+\frac{1}{3} a B c x^6+\frac{1}{7} A c^2 x^7+\frac{1}{8} B c^2 x^8\\ \end{align*}
Mathematica [A] time = 0.0028249, size = 65, normalized size = 1. \[ \frac{1}{3} a^2 A x^3+\frac{1}{4} a^2 B x^4+\frac{2}{5} a A c x^5+\frac{1}{3} a B c x^6+\frac{1}{7} A c^2 x^7+\frac{1}{8} B c^2 x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 54, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}A{x}^{3}}{3}}+{\frac{{a}^{2}B{x}^{4}}{4}}+{\frac{2\,aAc{x}^{5}}{5}}+{\frac{aBc{x}^{6}}{3}}+{\frac{A{c}^{2}{x}^{7}}{7}}+{\frac{B{c}^{2}{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990903, size = 72, normalized size = 1.11 \begin{align*} \frac{1}{8} \, B c^{2} x^{8} + \frac{1}{7} \, A c^{2} x^{7} + \frac{1}{3} \, B a c x^{6} + \frac{2}{5} \, A a c x^{5} + \frac{1}{4} \, B a^{2} x^{4} + \frac{1}{3} \, A a^{2} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33623, size = 128, normalized size = 1.97 \begin{align*} \frac{1}{8} x^{8} c^{2} B + \frac{1}{7} x^{7} c^{2} A + \frac{1}{3} x^{6} c a B + \frac{2}{5} x^{5} c a A + \frac{1}{4} x^{4} a^{2} B + \frac{1}{3} x^{3} a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.072835, size = 61, normalized size = 0.94 \begin{align*} \frac{A a^{2} x^{3}}{3} + \frac{2 A a c x^{5}}{5} + \frac{A c^{2} x^{7}}{7} + \frac{B a^{2} x^{4}}{4} + \frac{B a c x^{6}}{3} + \frac{B c^{2} x^{8}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11933, size = 72, normalized size = 1.11 \begin{align*} \frac{1}{8} \, B c^{2} x^{8} + \frac{1}{7} \, A c^{2} x^{7} + \frac{1}{3} \, B a c x^{6} + \frac{2}{5} \, A a c x^{5} + \frac{1}{4} \, B a^{2} x^{4} + \frac{1}{3} \, A a^{2} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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