Optimal. Leaf size=93 \[ \frac{3}{4} a^2 A c x^4+\frac{1}{2} a^3 A x^2+\frac{3}{5} a^2 B c x^5+\frac{1}{3} a^3 B x^3+\frac{1}{2} a A c^2 x^6+\frac{3}{7} a B c^2 x^7+\frac{1}{8} A c^3 x^8+\frac{1}{9} B c^3 x^9 \]
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Rubi [A] time = 0.0792587, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {766} \[ \frac{3}{4} a^2 A c x^4+\frac{1}{2} a^3 A x^2+\frac{3}{5} a^2 B c x^5+\frac{1}{3} a^3 B x^3+\frac{1}{2} a A c^2 x^6+\frac{3}{7} a B c^2 x^7+\frac{1}{8} A c^3 x^8+\frac{1}{9} B c^3 x^9 \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int x (A+B x) \left (a+c x^2\right )^3 \, dx &=\int \left (a^3 A x+a^3 B x^2+3 a^2 A c x^3+3 a^2 B c x^4+3 a A c^2 x^5+3 a B c^2 x^6+A c^3 x^7+B c^3 x^8\right ) \, dx\\ &=\frac{1}{2} a^3 A x^2+\frac{1}{3} a^3 B x^3+\frac{3}{4} a^2 A c x^4+\frac{3}{5} a^2 B c x^5+\frac{1}{2} a A c^2 x^6+\frac{3}{7} a B c^2 x^7+\frac{1}{8} A c^3 x^8+\frac{1}{9} B c^3 x^9\\ \end{align*}
Mathematica [A] time = 0.0027795, size = 93, normalized size = 1. \[ \frac{3}{4} a^2 A c x^4+\frac{1}{2} a^3 A x^2+\frac{3}{5} a^2 B c x^5+\frac{1}{3} a^3 B x^3+\frac{1}{2} a A c^2 x^6+\frac{3}{7} a B c^2 x^7+\frac{1}{8} A c^3 x^8+\frac{1}{9} B c^3 x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 78, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}A{x}^{2}}{2}}+{\frac{{a}^{3}B{x}^{3}}{3}}+{\frac{3\,{a}^{2}Ac{x}^{4}}{4}}+{\frac{3\,{a}^{2}Bc{x}^{5}}{5}}+{\frac{aA{c}^{2}{x}^{6}}{2}}+{\frac{3\,aB{c}^{2}{x}^{7}}{7}}+{\frac{A{c}^{3}{x}^{8}}{8}}+{\frac{B{c}^{3}{x}^{9}}{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02881, size = 104, normalized size = 1.12 \begin{align*} \frac{1}{9} \, B c^{3} x^{9} + \frac{1}{8} \, A c^{3} x^{8} + \frac{3}{7} \, B a c^{2} x^{7} + \frac{1}{2} \, A a c^{2} x^{6} + \frac{3}{5} \, B a^{2} c x^{5} + \frac{3}{4} \, A a^{2} c x^{4} + \frac{1}{3} \, B a^{3} x^{3} + \frac{1}{2} \, A a^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58022, size = 182, normalized size = 1.96 \begin{align*} \frac{1}{9} x^{9} c^{3} B + \frac{1}{8} x^{8} c^{3} A + \frac{3}{7} x^{7} c^{2} a B + \frac{1}{2} x^{6} c^{2} a A + \frac{3}{5} x^{5} c a^{2} B + \frac{3}{4} x^{4} c a^{2} A + \frac{1}{3} x^{3} a^{3} B + \frac{1}{2} x^{2} a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.083398, size = 92, normalized size = 0.99 \begin{align*} \frac{A a^{3} x^{2}}{2} + \frac{3 A a^{2} c x^{4}}{4} + \frac{A a c^{2} x^{6}}{2} + \frac{A c^{3} x^{8}}{8} + \frac{B a^{3} x^{3}}{3} + \frac{3 B a^{2} c x^{5}}{5} + \frac{3 B a c^{2} x^{7}}{7} + \frac{B c^{3} x^{9}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11881, size = 104, normalized size = 1.12 \begin{align*} \frac{1}{9} \, B c^{3} x^{9} + \frac{1}{8} \, A c^{3} x^{8} + \frac{3}{7} \, B a c^{2} x^{7} + \frac{1}{2} \, A a c^{2} x^{6} + \frac{3}{5} \, B a^{2} c x^{5} + \frac{3}{4} \, A a^{2} c x^{4} + \frac{1}{3} \, B a^{3} x^{3} + \frac{1}{2} \, A a^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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