Optimal. Leaf size=87 \[ \frac{1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{1}{5} a^2 c^2 x^5+\frac{2}{11} b d x^{11} (a d+b c)+\frac{2}{7} a c x^7 (a d+b c)+\frac{1}{13} b^2 d^2 x^{13} \]
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Rubi [A] time = 0.0618824, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{1}{5} a^2 c^2 x^5+\frac{2}{11} b d x^{11} (a d+b c)+\frac{2}{7} a c x^7 (a d+b c)+\frac{1}{13} b^2 d^2 x^{13} \]
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 )^2 \, dx &=\int \left (a^2 c^2 x^4+2 a c (b c+a d) x^6+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^8+2 b d (b c+a d) x^{10}+b^2 d^2 x^{12}\right ) \, dx\\ &=\frac{1}{5} a^2 c^2 x^5+\frac{2}{7} a c (b c+a d) x^7+\frac{1}{9} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^9+\frac{2}{11} b d (b c+a d) x^{11}+\frac{1}{13} b^2 d^2 x^{13}\\ \end{align*}
Mathematica [A] time = 0.0154137, size = 87, normalized size = 1. \[ \frac{1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{1}{5} a^2 c^2 x^5+\frac{2}{11} b d x^{11} (a d+b c)+\frac{2}{7} a c x^7 (a d+b c)+\frac{1}{13} b^2 d^2 x^{13} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 90, normalized size = 1. \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{13}}{13}}+{\frac{ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){x}^{11}}{11}}+{\frac{ \left ({a}^{2}{d}^{2}+4\,cabd+{b}^{2}{c}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){x}^{7}}{7}}+{\frac{{a}^{2}{c}^{2}{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99778, size = 115, normalized size = 1.32 \begin{align*} \frac{1}{13} \, b^{2} d^{2} x^{13} + \frac{2}{11} \,{\left (b^{2} c d + a b d^{2}\right )} x^{11} + \frac{1}{9} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{9} + \frac{1}{5} \, a^{2} c^{2} x^{5} + \frac{2}{7} \,{\left (a b c^{2} + a^{2} c d\right )} x^{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.0882, size = 225, normalized size = 2.59 \begin{align*} \frac{1}{13} x^{13} d^{2} b^{2} + \frac{2}{11} x^{11} d c b^{2} + \frac{2}{11} x^{11} d^{2} b a + \frac{1}{9} x^{9} c^{2} b^{2} + \frac{4}{9} x^{9} d c b a + \frac{1}{9} x^{9} d^{2} a^{2} + \frac{2}{7} x^{7} c^{2} b a + \frac{2}{7} x^{7} d c a^{2} + \frac{1}{5} x^{5} c^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.078768, size = 100, normalized size = 1.15 \begin{align*} \frac{a^{2} c^{2} x^{5}}{5} + \frac{b^{2} d^{2} x^{13}}{13} + x^{11} \left (\frac{2 a b d^{2}}{11} + \frac{2 b^{2} c d}{11}\right ) + x^{9} \left (\frac{a^{2} d^{2}}{9} + \frac{4 a b c d}{9} + \frac{b^{2} c^{2}}{9}\right ) + x^{7} \left (\frac{2 a^{2} c d}{7} + \frac{2 a b c^{2}}{7}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13503, size = 127, normalized size = 1.46 \begin{align*} \frac{1}{13} \, b^{2} d^{2} x^{13} + \frac{2}{11} \, b^{2} c d x^{11} + \frac{2}{11} \, a b d^{2} x^{11} + \frac{1}{9} \, b^{2} c^{2} x^{9} + \frac{4}{9} \, a b c d x^{9} + \frac{1}{9} \, a^{2} d^{2} x^{9} + \frac{2}{7} \, a b c^{2} x^{7} + \frac{2}{7} \, a^{2} c d x^{7} + \frac{1}{5} \, a^{2} c^{2} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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