Optimal. Leaf size=55 \[ \frac{1}{3} a^2 A x^3+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{9} b^2 B x^9 \]
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Rubi [A] time = 0.0361749, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ \frac{1}{3} a^2 A x^3+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{9} b^2 B x^9 \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx &=\int \left (a^2 A x^2+a (2 A b+a B) x^4+b (A b+2 a B) x^6+b^2 B x^8\right ) \, dx\\ &=\frac{1}{3} a^2 A x^3+\frac{1}{5} a (2 A b+a B) x^5+\frac{1}{7} b (A b+2 a B) x^7+\frac{1}{9} b^2 B x^9\\ \end{align*}
Mathematica [A] time = 0.0077699, size = 55, normalized size = 1. \[ \frac{1}{3} a^2 A x^3+\frac{1}{7} b x^7 (2 a B+A b)+\frac{1}{5} a x^5 (a B+2 A b)+\frac{1}{9} b^2 B x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 52, normalized size = 1. \begin{align*}{\frac{{b}^{2}B{x}^{9}}{9}}+{\frac{ \left ({b}^{2}A+2\,abB \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,abA+{a}^{2}B \right ){x}^{5}}{5}}+{\frac{{a}^{2}A{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02242, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{9} \, B b^{2} x^{9} + \frac{1}{7} \,{\left (2 \, B a b + A b^{2}\right )} x^{7} + \frac{1}{3} \, A a^{2} x^{3} + \frac{1}{5} \,{\left (B a^{2} + 2 \, A a b\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25053, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{9} x^{9} b^{2} B + \frac{2}{7} x^{7} b a B + \frac{1}{7} x^{7} b^{2} A + \frac{1}{5} x^{5} a^{2} B + \frac{2}{5} x^{5} b a A + \frac{1}{3} x^{3} a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.065141, size = 56, normalized size = 1.02 \begin{align*} \frac{A a^{2} x^{3}}{3} + \frac{B b^{2} x^{9}}{9} + x^{7} \left (\frac{A b^{2}}{7} + \frac{2 B a b}{7}\right ) + x^{5} \left (\frac{2 A a b}{5} + \frac{B a^{2}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11797, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{9} \, B b^{2} x^{9} + \frac{2}{7} \, B a b x^{7} + \frac{1}{7} \, A b^{2} x^{7} + \frac{1}{5} \, B a^{2} x^{5} + \frac{2}{5} \, A a b x^{5} + \frac{1}{3} \, A a^{2} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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