Optimal. Leaf size=55 \[ \frac{1}{3} a^2 c x^3+\frac{1}{7} b x^7 (2 a d+b c)+\frac{1}{5} a x^5 (a d+2 b c)+\frac{1}{9} b^2 d x^9 \]
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Rubi [A] time = 0.0296691, 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} \[ \frac{1}{3} a^2 c x^3+\frac{1}{7} b x^7 (2 a d+b c)+\frac{1}{5} a x^5 (a d+2 b c)+\frac{1}{9} b^2 d x^9 \]
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 (c+d x^2\right ) \, dx &=\int \left (a^2 c x^2+a (2 b c+a d) x^4+b (b c+2 a d) x^6+b^2 d x^8\right ) \, dx\\ &=\frac{1}{3} a^2 c x^3+\frac{1}{5} a (2 b c+a d) x^5+\frac{1}{7} b (b c+2 a d) x^7+\frac{1}{9} b^2 d x^9\\ \end{align*}
Mathematica [A] time = 0.0074007, size = 55, normalized size = 1. \[ \frac{1}{3} a^2 c x^3+\frac{1}{7} b x^7 (2 a d+b c)+\frac{1}{5} a x^5 (a d+2 b c)+\frac{1}{9} b^2 d x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 52, normalized size = 1. \begin{align*}{\frac{{b}^{2}d{x}^{9}}{9}}+{\frac{ \left ( 2\,abd+{b}^{2}c \right ){x}^{7}}{7}}+{\frac{ \left ({a}^{2}d+2\,abc \right ){x}^{5}}{5}}+{\frac{{a}^{2}c{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00241, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{9} \, b^{2} d x^{9} + \frac{1}{7} \,{\left (b^{2} c + 2 \, a b d\right )} x^{7} + \frac{1}{3} \, a^{2} c x^{3} + \frac{1}{5} \,{\left (2 \, a b c + a^{2} d\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.1323, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{9} x^{9} d b^{2} + \frac{1}{7} x^{7} c b^{2} + \frac{2}{7} x^{7} d b a + \frac{2}{5} x^{5} c b a + \frac{1}{5} x^{5} d a^{2} + \frac{1}{3} x^{3} c a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.067026, size = 56, normalized size = 1.02 \begin{align*} \frac{a^{2} c x^{3}}{3} + \frac{b^{2} d x^{9}}{9} + x^{7} \left (\frac{2 a b d}{7} + \frac{b^{2} c}{7}\right ) + x^{5} \left (\frac{a^{2} d}{5} + \frac{2 a b c}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18801, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{9} \, b^{2} d x^{9} + \frac{1}{7} \, b^{2} c x^{7} + \frac{2}{7} \, a b d x^{7} + \frac{2}{5} \, a b c x^{5} + \frac{1}{5} \, a^{2} d x^{5} + \frac{1}{3} \, a^{2} c x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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