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 = 0.0346943, 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} \[ \frac{1}{5} a^2 c x^5+\frac{1}{9} b x^9 (2 a d+b c)+\frac{1}{7} a x^7 (a d+2 b c)+\frac{1}{11} b^2 d x^{11} \]
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*}
Mathematica [A] time = 0.0089776, size = 55, normalized size = 1. \[ \frac{1}{5} a^2 c x^5+\frac{1}{9} b x^9 (2 a d+b c)+\frac{1}{7} a x^7 (a d+2 b c)+\frac{1}{11} b^2 d x^{11} \]
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}^{11}}{11}}+{\frac{ \left ( 2\,abd+{b}^{2}c \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{2}d+2\,abc \right ){x}^{7}}{7}}+{\frac{{a}^{2}c{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01054, size = 69, normalized size = 1.25 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.1569, size = 131, normalized size = 2.38 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.067704, size = 56, normalized size = 1.02 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14615, size = 72, normalized size = 1.31 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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