Optimal. Leaf size=74 \[ \frac {1}{3} a A x^3+\frac {1}{4} a B x^4+\frac {1}{5} (A b+a C) x^5+\frac {1}{6} b B x^6+\frac {1}{7} (A c+b C) x^7+\frac {1}{8} B c x^8+\frac {1}{9} c C x^9 \]
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Rubi [A]
time = 0.05, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1642}
\begin {gather*} \frac {1}{5} x^5 (a C+A b)+\frac {1}{3} a A x^3+\frac {1}{4} a B x^4+\frac {1}{7} x^7 (A c+b C)+\frac {1}{6} b B x^6+\frac {1}{8} B c x^8+\frac {1}{9} c C x^9 \end {gather*}
Antiderivative was successfully verified.
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Rule 1642
Rubi steps
\begin {align*} \int x^2 \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a A x^2+a B x^3+(A b+a C) x^4+b B x^5+(A c+b C) x^6+B c x^7+c C x^8\right ) \, dx\\ &=\frac {1}{3} a A x^3+\frac {1}{4} a B x^4+\frac {1}{5} (A b+a C) x^5+\frac {1}{6} b B x^6+\frac {1}{7} (A c+b C) x^7+\frac {1}{8} B c x^8+\frac {1}{9} c C x^9\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 74, normalized size = 1.00 \begin {gather*} \frac {1}{3} a A x^3+\frac {1}{4} a B x^4+\frac {1}{5} (A b+a C) x^5+\frac {1}{6} b B x^6+\frac {1}{7} (A c+b C) x^7+\frac {1}{8} B c x^8+\frac {1}{9} c C x^9 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 61, normalized size = 0.82
method | result | size |
default | \(\frac {a A \,x^{3}}{3}+\frac {a B \,x^{4}}{4}+\frac {\left (A b +a C \right ) x^{5}}{5}+\frac {b B \,x^{6}}{6}+\frac {\left (A c +b C \right ) x^{7}}{7}+\frac {B c \,x^{8}}{8}+\frac {c C \,x^{9}}{9}\) | \(61\) |
norman | \(\frac {c C \,x^{9}}{9}+\frac {B c \,x^{8}}{8}+\left (\frac {A c}{7}+\frac {b C}{7}\right ) x^{7}+\frac {b B \,x^{6}}{6}+\left (\frac {A b}{5}+\frac {a C}{5}\right ) x^{5}+\frac {a B \,x^{4}}{4}+\frac {a A \,x^{3}}{3}\) | \(63\) |
gosper | \(\frac {1}{9} c C \,x^{9}+\frac {1}{8} B c \,x^{8}+\frac {1}{7} x^{7} A c +\frac {1}{7} x^{7} b C +\frac {1}{6} b B \,x^{6}+\frac {1}{5} x^{5} A b +\frac {1}{5} x^{5} a C +\frac {1}{4} a B \,x^{4}+\frac {1}{3} a A \,x^{3}\) | \(65\) |
risch | \(\frac {1}{9} c C \,x^{9}+\frac {1}{8} B c \,x^{8}+\frac {1}{7} x^{7} A c +\frac {1}{7} x^{7} b C +\frac {1}{6} b B \,x^{6}+\frac {1}{5} x^{5} A b +\frac {1}{5} x^{5} a C +\frac {1}{4} a B \,x^{4}+\frac {1}{3} a A \,x^{3}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 60, normalized size = 0.81 \begin {gather*} \frac {1}{9} \, C c x^{9} + \frac {1}{8} \, B c x^{8} + \frac {1}{6} \, B b x^{6} + \frac {1}{7} \, {\left (C b + A c\right )} x^{7} + \frac {1}{4} \, B a x^{4} + \frac {1}{5} \, {\left (C a + A b\right )} x^{5} + \frac {1}{3} \, A a x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 60, normalized size = 0.81 \begin {gather*} \frac {1}{9} \, C c x^{9} + \frac {1}{8} \, B c x^{8} + \frac {1}{6} \, B b x^{6} + \frac {1}{7} \, {\left (C b + A c\right )} x^{7} + \frac {1}{4} \, B a x^{4} + \frac {1}{5} \, {\left (C a + A b\right )} x^{5} + \frac {1}{3} \, A a x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 68, normalized size = 0.92 \begin {gather*} \frac {A a x^{3}}{3} + \frac {B a x^{4}}{4} + \frac {B b x^{6}}{6} + \frac {B c x^{8}}{8} + \frac {C c x^{9}}{9} + x^{7} \left (\frac {A c}{7} + \frac {C b}{7}\right ) + x^{5} \left (\frac {A b}{5} + \frac {C a}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.79, size = 64, normalized size = 0.86 \begin {gather*} \frac {1}{9} \, C c x^{9} + \frac {1}{8} \, B c x^{8} + \frac {1}{7} \, C b x^{7} + \frac {1}{7} \, A c x^{7} + \frac {1}{6} \, B b x^{6} + \frac {1}{5} \, C a x^{5} + \frac {1}{5} \, A b x^{5} + \frac {1}{4} \, B a x^{4} + \frac {1}{3} \, A a x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 62, normalized size = 0.84 \begin {gather*} \frac {C\,c\,x^9}{9}+\frac {B\,c\,x^8}{8}+\left (\frac {A\,c}{7}+\frac {C\,b}{7}\right )\,x^7+\frac {B\,b\,x^6}{6}+\left (\frac {A\,b}{5}+\frac {C\,a}{5}\right )\,x^5+\frac {B\,a\,x^4}{4}+\frac {A\,a\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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