Optimal. Leaf size=74 \[ \frac {1}{4} x^4 (a C+A b)+\frac {1}{2} a A x^2+\frac {1}{3} a B x^3+\frac {1}{6} x^6 (A c+b C)+\frac {1}{5} b B x^5+\frac {1}{7} B c x^7+\frac {1}{8} c C x^8 \]
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Rubi [A] time = 0.06, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1628} \[ \frac {1}{4} x^4 (a C+A b)+\frac {1}{2} a A x^2+\frac {1}{3} a B x^3+\frac {1}{6} x^6 (A c+b C)+\frac {1}{5} b B x^5+\frac {1}{7} B c x^7+\frac {1}{8} c C x^8 \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin {align*} \int x \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a A x+a B x^2+(A b+a C) x^3+b B x^4+(A c+b C) x^5+B c x^6+c C x^7\right ) \, dx\\ &=\frac {1}{2} a A x^2+\frac {1}{3} a B x^3+\frac {1}{4} (A b+a C) x^4+\frac {1}{5} b B x^5+\frac {1}{6} (A c+b C) x^6+\frac {1}{7} B c x^7+\frac {1}{8} c C x^8\\ \end {align*}
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Mathematica [A] time = 0.01, size = 74, normalized size = 1.00 \[ \frac {1}{4} x^4 (a C+A b)+\frac {1}{2} a A x^2+\frac {1}{3} a B x^3+\frac {1}{6} x^6 (A c+b C)+\frac {1}{5} b B x^5+\frac {1}{7} B c x^7+\frac {1}{8} c C x^8 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 64, normalized size = 0.86 \[ \frac {1}{8} x^{8} c C + \frac {1}{7} x^{7} c B + \frac {1}{6} x^{6} b C + \frac {1}{6} x^{6} c A + \frac {1}{5} x^{5} b B + \frac {1}{4} x^{4} a C + \frac {1}{4} x^{4} b A + \frac {1}{3} x^{3} a B + \frac {1}{2} x^{2} a A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 64, normalized size = 0.86 \[ \frac {1}{8} \, C c x^{8} + \frac {1}{7} \, B c x^{7} + \frac {1}{6} \, C b x^{6} + \frac {1}{6} \, A c x^{6} + \frac {1}{5} \, B b x^{5} + \frac {1}{4} \, C a x^{4} + \frac {1}{4} \, A b x^{4} + \frac {1}{3} \, B a x^{3} + \frac {1}{2} \, A a x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 61, normalized size = 0.82 \[ \frac {C c \,x^{8}}{8}+\frac {B c \,x^{7}}{7}+\frac {B b \,x^{5}}{5}+\frac {\left (A c +b C \right ) x^{6}}{6}+\frac {B a \,x^{3}}{3}+\frac {A a \,x^{2}}{2}+\frac {\left (A b +a C \right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 60, normalized size = 0.81 \[ \frac {1}{8} \, C c x^{8} + \frac {1}{7} \, B c x^{7} + \frac {1}{5} \, B b x^{5} + \frac {1}{6} \, {\left (C b + A c\right )} x^{6} + \frac {1}{3} \, B a x^{3} + \frac {1}{4} \, {\left (C a + A b\right )} x^{4} + \frac {1}{2} \, A a x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 62, normalized size = 0.84 \[ \frac {C\,c\,x^8}{8}+\frac {B\,c\,x^7}{7}+\left (\frac {A\,c}{6}+\frac {C\,b}{6}\right )\,x^6+\frac {B\,b\,x^5}{5}+\left (\frac {A\,b}{4}+\frac {C\,a}{4}\right )\,x^4+\frac {B\,a\,x^3}{3}+\frac {A\,a\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 68, normalized size = 0.92 \[ \frac {A a x^{2}}{2} + \frac {B a x^{3}}{3} + \frac {B b x^{5}}{5} + \frac {B c x^{7}}{7} + \frac {C c x^{8}}{8} + x^{6} \left (\frac {A c}{6} + \frac {C b}{6}\right ) + x^{4} \left (\frac {A b}{4} + \frac {C a}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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