Optimal. Leaf size=93 \[ \frac {1}{2} a^3 A x^2+\frac {1}{3} a^3 B x^3+\frac {3}{4} a^2 A c x^4+\frac {3}{5} a^2 B c x^5+\frac {1}{2} a A c^2 x^6+\frac {3}{7} a B c^2 x^7+\frac {1}{8} A c^3 x^8+\frac {1}{9} B c^3 x^9 \]
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Rubi [A] time = 0.08, antiderivative size = 93, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.00, size = 93, normalized size = 1.00 \[ \frac {1}{2} a^3 A x^2+\frac {1}{3} a^3 B x^3+\frac {3}{4} a^2 A c x^4+\frac {3}{5} a^2 B c x^5+\frac {1}{2} a A c^2 x^6+\frac {3}{7} a B c^2 x^7+\frac {1}{8} A c^3 x^8+\frac {1}{9} B c^3 x^9 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 77, normalized size = 0.83 \[ \frac {1}{9} x^{9} c^{3} B + \frac {1}{8} x^{8} c^{3} A + \frac {3}{7} x^{7} c^{2} a B + \frac {1}{2} x^{6} c^{2} a A + \frac {3}{5} x^{5} c a^{2} B + \frac {3}{4} x^{4} c a^{2} A + \frac {1}{3} x^{3} a^{3} B + \frac {1}{2} x^{2} a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 77, normalized size = 0.83 \[ \frac {1}{9} \, B c^{3} x^{9} + \frac {1}{8} \, A c^{3} x^{8} + \frac {3}{7} \, B a c^{2} x^{7} + \frac {1}{2} \, A a c^{2} x^{6} + \frac {3}{5} \, B a^{2} c x^{5} + \frac {3}{4} \, A a^{2} c x^{4} + \frac {1}{3} \, B a^{3} x^{3} + \frac {1}{2} \, A a^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 78, normalized size = 0.84 \[ \frac {1}{9} B \,c^{3} x^{9}+\frac {1}{8} A \,c^{3} x^{8}+\frac {3}{7} B a \,c^{2} x^{7}+\frac {1}{2} A a \,c^{2} x^{6}+\frac {3}{5} B \,a^{2} c \,x^{5}+\frac {3}{4} A \,a^{2} c \,x^{4}+\frac {1}{3} B \,a^{3} x^{3}+\frac {1}{2} A \,a^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 77, normalized size = 0.83 \[ \frac {1}{9} \, B c^{3} x^{9} + \frac {1}{8} \, A c^{3} x^{8} + \frac {3}{7} \, B a c^{2} x^{7} + \frac {1}{2} \, A a c^{2} x^{6} + \frac {3}{5} \, B a^{2} c x^{5} + \frac {3}{4} \, A a^{2} c x^{4} + \frac {1}{3} \, B a^{3} x^{3} + \frac {1}{2} \, A a^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 77, normalized size = 0.83 \[ \frac {B\,a^3\,x^3}{3}+\frac {A\,a^3\,x^2}{2}+\frac {3\,B\,a^2\,c\,x^5}{5}+\frac {3\,A\,a^2\,c\,x^4}{4}+\frac {3\,B\,a\,c^2\,x^7}{7}+\frac {A\,a\,c^2\,x^6}{2}+\frac {B\,c^3\,x^9}{9}+\frac {A\,c^3\,x^8}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 92, normalized size = 0.99 \[ \frac {A a^{3} x^{2}}{2} + \frac {3 A a^{2} c x^{4}}{4} + \frac {A a c^{2} x^{6}}{2} + \frac {A c^{3} x^{8}}{8} + \frac {B a^{3} x^{3}}{3} + \frac {3 B a^{2} c x^{5}}{5} + \frac {3 B a c^{2} x^{7}}{7} + \frac {B c^{3} x^{9}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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