Optimal. Leaf size=55 \[ \frac {1}{5} a^2 c x^5+\frac {1}{9} b x^9 (2 a d+b c)+\frac {1}{7} a x^7 (a d+2 b c)+\frac {1}{11} b^2 d x^{11} \]
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Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \begin {gather*} \frac {1}{5} a^2 c x^5+\frac {1}{9} b x^9 (2 a d+b c)+\frac {1}{7} a x^7 (a d+2 b c)+\frac {1}{11} b^2 d x^{11} \end {gather*}
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\int \left (a^2 c x^4+a (2 b c+a d) x^6+b (b c+2 a d) x^8+b^2 d x^{10}\right ) \, dx\\ &=\frac {1}{5} a^2 c x^5+\frac {1}{7} a (2 b c+a d) x^7+\frac {1}{9} b (b c+2 a d) x^9+\frac {1}{11} b^2 d x^{11}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 55, normalized size = 1.00 \begin {gather*} \frac {1}{5} a^2 c x^5+\frac {1}{9} b x^9 (2 a d+b c)+\frac {1}{7} a x^7 (a d+2 b c)+\frac {1}{11} b^2 d x^{11} \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^4 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.50, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{11} x^{11} d b^{2} + \frac {1}{9} x^{9} c b^{2} + \frac {2}{9} x^{9} d b a + \frac {2}{7} x^{7} c b a + \frac {1}{7} x^{7} d a^{2} + \frac {1}{5} x^{5} c a^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{11} \, b^{2} d x^{11} + \frac {1}{9} \, b^{2} c x^{9} + \frac {2}{9} \, a b d x^{9} + \frac {2}{7} \, a b c x^{7} + \frac {1}{7} \, a^{2} d x^{7} + \frac {1}{5} \, a^{2} c x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 52, normalized size = 0.95 \begin {gather*} \frac {b^{2} d \,x^{11}}{11}+\frac {\left (2 a b d +b^{2} c \right ) x^{9}}{9}+\frac {a^{2} c \,x^{5}}{5}+\frac {\left (a^{2} d +2 a b c \right ) x^{7}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{11} \, b^{2} d x^{11} + \frac {1}{9} \, {\left (b^{2} c + 2 \, a b d\right )} x^{9} + \frac {1}{5} \, a^{2} c x^{5} + \frac {1}{7} \, {\left (2 \, a b c + a^{2} d\right )} x^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 51, normalized size = 0.93 \begin {gather*} x^7\,\left (\frac {d\,a^2}{7}+\frac {2\,b\,c\,a}{7}\right )+x^9\,\left (\frac {c\,b^2}{9}+\frac {2\,a\,d\,b}{9}\right )+\frac {a^2\,c\,x^5}{5}+\frac {b^2\,d\,x^{11}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 56, normalized size = 1.02 \begin {gather*} \frac {a^{2} c x^{5}}{5} + \frac {b^{2} d x^{11}}{11} + x^{9} \left (\frac {2 a b d}{9} + \frac {b^{2} c}{9}\right ) + x^{7} \left (\frac {a^{2} d}{7} + \frac {2 a b c}{7}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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