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 = 0.0475207, antiderivative size = 55, normalized size of antiderivative = 1., 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 1593
Rule 448
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*}
Mathematica [A] time = 0.0096894, size = 55, normalized size = 1. \[ \frac{1}{5} A b^2 x^5+\frac{1}{9} c x^9 (A c+2 b B)+\frac{1}{7} b x^7 (2 A c+b B)+\frac{1}{11} B c^2 x^{11} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 52, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{11}}{11}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,Abc+B{b}^{2} \right ){x}^{7}}{7}}+{\frac{A{b}^{2}{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1433, size = 69, normalized size = 1.25 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.41931, size = 131, normalized size = 2.38 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.070626, size = 56, normalized size = 1.02 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20551, size = 72, normalized size = 1.31 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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