Optimal. Leaf size=55 \[ \frac {1}{5} A b^2 x^5+\frac {1}{9} c x^9 (A c+2 b B)+\frac {1}{7} b x^7 (2 A c+b B)+\frac {1}{11} B c^2 x^{11} \]
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Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1593, 448} \[ \frac {1}{5} A b^2 x^5+\frac {1}{9} c x^9 (A c+2 b B)+\frac {1}{7} b x^7 (2 A c+b B)+\frac {1}{11} B c^2 x^{11} \]
Antiderivative was successfully verified.
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Rule 448
Rule 1593
Rubi steps
\begin {align*} \int \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx &=\int x^4 \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^4+b (b B+2 A c) x^6+c (2 b B+A c) x^8+B c^2 x^{10}\right ) \, dx\\ &=\frac {1}{5} A b^2 x^5+\frac {1}{7} b (b B+2 A c) x^7+\frac {1}{9} c (2 b B+A c) x^9+\frac {1}{11} B c^2 x^{11}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 55, normalized size = 1.00 \[ \frac {1}{5} A b^2 x^5+\frac {1}{9} c x^9 (A c+2 b B)+\frac {1}{7} b x^7 (2 A c+b B)+\frac {1}{11} B c^2 x^{11} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 53, normalized size = 0.96 \[ \frac {1}{11} x^{11} c^{2} B + \frac {2}{9} x^{9} c b B + \frac {1}{9} x^{9} c^{2} A + \frac {1}{7} x^{7} b^{2} B + \frac {2}{7} x^{7} c b A + \frac {1}{5} x^{5} b^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 0.96 \[ \frac {1}{11} \, B c^{2} x^{11} + \frac {2}{9} \, B b c x^{9} + \frac {1}{9} \, A c^{2} x^{9} + \frac {1}{7} \, B b^{2} x^{7} + \frac {2}{7} \, A b c x^{7} + \frac {1}{5} \, A b^{2} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 52, normalized size = 0.95 \[ \frac {B \,c^{2} x^{11}}{11}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{9}}{9}+\frac {A \,b^{2} x^{5}}{5}+\frac {\left (2 A b c +B \,b^{2}\right ) x^{7}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 51, normalized size = 0.93 \[ \frac {1}{11} \, B c^{2} x^{11} + \frac {1}{9} \, {\left (2 \, B b c + A c^{2}\right )} x^{9} + \frac {1}{5} \, A b^{2} x^{5} + \frac {1}{7} \, {\left (B b^{2} + 2 \, A b c\right )} x^{7} \]
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 \[ x^7\,\left (\frac {B\,b^2}{7}+\frac {2\,A\,c\,b}{7}\right )+x^9\,\left (\frac {A\,c^2}{9}+\frac {2\,B\,b\,c}{9}\right )+\frac {A\,b^2\,x^5}{5}+\frac {B\,c^2\,x^{11}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 56, normalized size = 1.02 \[ \frac {A b^{2} x^{5}}{5} + \frac {B c^{2} x^{11}}{11} + x^{9} \left (\frac {A c^{2}}{9} + \frac {2 B b c}{9}\right ) + x^{7} \left (\frac {2 A b c}{7} + \frac {B b^{2}}{7}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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