Optimal. Leaf size=65 \[ \frac {1}{2} a^2 A x^2+\frac {1}{3} a^2 B x^3+\frac {1}{2} a A c x^4+\frac {2}{5} a B c x^5+\frac {1}{6} A c^2 x^6+\frac {1}{7} B c^2 x^7 \]
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Rubi [A] time = 0.04, antiderivative size = 65, 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} \begin {gather*} \frac {1}{2} a^2 A x^2+\frac {1}{3} a^2 B x^3+\frac {1}{2} a A c x^4+\frac {2}{5} a B c x^5+\frac {1}{6} A c^2 x^6+\frac {1}{7} B c^2 x^7 \end {gather*}
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 )^2 \, dx &=\int \left (a^2 A x+a^2 B x^2+2 a A c x^3+2 a B c x^4+A c^2 x^5+B c^2 x^6\right ) \, dx\\ &=\frac {1}{2} a^2 A x^2+\frac {1}{3} a^2 B x^3+\frac {1}{2} a A c x^4+\frac {2}{5} a B c x^5+\frac {1}{6} A c^2 x^6+\frac {1}{7} B c^2 x^7\\ \end {align*}
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Mathematica [A] time = 0.00, size = 65, normalized size = 1.00 \begin {gather*} \frac {1}{2} a^2 A x^2+\frac {1}{3} a^2 B x^3+\frac {1}{2} a A c x^4+\frac {2}{5} a B c x^5+\frac {1}{6} A c^2 x^6+\frac {1}{7} B c^2 x^7 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x (A+B x) \left (a+c x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.36, size = 53, normalized size = 0.82 \begin {gather*} \frac {1}{7} x^{7} c^{2} B + \frac {1}{6} x^{6} c^{2} A + \frac {2}{5} x^{5} c a B + \frac {1}{2} x^{4} c a A + \frac {1}{3} x^{3} a^{2} B + \frac {1}{2} x^{2} a^{2} A \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 0.82 \begin {gather*} \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{6} \, A c^{2} x^{6} + \frac {2}{5} \, B a c x^{5} + \frac {1}{2} \, A a c x^{4} + \frac {1}{3} \, B a^{2} x^{3} + \frac {1}{2} \, A a^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 54, normalized size = 0.83 \begin {gather*} \frac {1}{7} B \,c^{2} x^{7}+\frac {1}{6} A \,c^{2} x^{6}+\frac {2}{5} B a c \,x^{5}+\frac {1}{2} A a c \,x^{4}+\frac {1}{3} B \,a^{2} x^{3}+\frac {1}{2} A \,a^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 53, normalized size = 0.82 \begin {gather*} \frac {1}{7} \, B c^{2} x^{7} + \frac {1}{6} \, A c^{2} x^{6} + \frac {2}{5} \, B a c x^{5} + \frac {1}{2} \, A a c x^{4} + \frac {1}{3} \, B a^{2} x^{3} + \frac {1}{2} \, A a^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 53, normalized size = 0.82 \begin {gather*} \frac {B\,a^2\,x^3}{3}+\frac {A\,a^2\,x^2}{2}+\frac {2\,B\,a\,c\,x^5}{5}+\frac {A\,a\,c\,x^4}{2}+\frac {B\,c^2\,x^7}{7}+\frac {A\,c^2\,x^6}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 61, normalized size = 0.94 \begin {gather*} \frac {A a^{2} x^{2}}{2} + \frac {A a c x^{4}}{2} + \frac {A c^{2} x^{6}}{6} + \frac {B a^{2} x^{3}}{3} + \frac {2 B a c x^{5}}{5} + \frac {B c^{2} x^{7}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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