Optimal. Leaf size=45 \[ a^2 A x+\frac{2}{3} a A c x^3+\frac{B \left (a+c x^2\right )^3}{6 c}+\frac{1}{5} A c^2 x^5 \]
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Rubi [A] time = 0.0148019, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {641, 194} \[ a^2 A x+\frac{2}{3} a A c x^3+\frac{B \left (a+c x^2\right )^3}{6 c}+\frac{1}{5} A c^2 x^5 \]
Antiderivative was successfully verified.
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Rule 641
Rule 194
Rubi steps
\begin{align*} \int (A+B x) \left (a+c x^2\right )^2 \, dx &=\frac{B \left (a+c x^2\right )^3}{6 c}+A \int \left (a+c x^2\right )^2 \, dx\\ &=\frac{B \left (a+c x^2\right )^3}{6 c}+A \int \left (a^2+2 a c x^2+c^2 x^4\right ) \, dx\\ &=a^2 A x+\frac{2}{3} a A c x^3+\frac{1}{5} A c^2 x^5+\frac{B \left (a+c x^2\right )^3}{6 c}\\ \end{align*}
Mathematica [A] time = 0.0020613, size = 60, normalized size = 1.33 \[ a^2 A x+\frac{1}{2} a^2 B x^2+\frac{2}{3} a A c x^3+\frac{1}{2} a B c x^4+\frac{1}{5} A c^2 x^5+\frac{1}{6} B c^2 x^6 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 51, normalized size = 1.1 \begin{align*}{\frac{B{c}^{2}{x}^{6}}{6}}+{\frac{A{c}^{2}{x}^{5}}{5}}+{\frac{aBc{x}^{4}}{2}}+{\frac{2\,aAc{x}^{3}}{3}}+{\frac{{a}^{2}B{x}^{2}}{2}}+{a}^{2}Ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03248, size = 68, normalized size = 1.51 \begin{align*} \frac{1}{6} \, B c^{2} x^{6} + \frac{1}{5} \, A c^{2} x^{5} + \frac{1}{2} \, B a c x^{4} + \frac{2}{3} \, A a c x^{3} + \frac{1}{2} \, B a^{2} x^{2} + A a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28154, size = 120, normalized size = 2.67 \begin{align*} \frac{1}{6} x^{6} c^{2} B + \frac{1}{5} x^{5} c^{2} A + \frac{1}{2} x^{4} c a B + \frac{2}{3} x^{3} c a A + \frac{1}{2} x^{2} a^{2} B + x a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.12706, size = 58, normalized size = 1.29 \begin{align*} A a^{2} x + \frac{2 A a c x^{3}}{3} + \frac{A c^{2} x^{5}}{5} + \frac{B a^{2} x^{2}}{2} + \frac{B a c x^{4}}{2} + \frac{B c^{2} x^{6}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10761, size = 68, normalized size = 1.51 \begin{align*} \frac{1}{6} \, B c^{2} x^{6} + \frac{1}{5} \, A c^{2} x^{5} + \frac{1}{2} \, B a c x^{4} + \frac{2}{3} \, A a c x^{3} + \frac{1}{2} \, B a^{2} x^{2} + A a^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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