Optimal. Leaf size=82 \[ \frac{1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{13} b d x^{13} (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{17} b^2 d^2 x^{17} \]
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Rubi [A] time = 0.0491069, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {373} \[ \frac{1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{13} b d x^{13} (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{17} b^2 d^2 x^{17} \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin{align*} \int \left (a+b x^4\right )^2 \left (c+d x^4\right )^2 \, dx &=\int \left (a^2 c^2+2 a c (b c+a d) x^4+\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^{12}+b^2 d^2 x^{16}\right ) \, dx\\ &=a^2 c^2 x+\frac{2}{5} a c (b c+a d) x^5+\frac{1}{9} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^9+\frac{2}{13} b d (b c+a d) x^{13}+\frac{1}{17} b^2 d^2 x^{17}\\ \end{align*}
Mathematica [A] time = 0.0166878, size = 82, normalized size = 1. \[ \frac{1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{13} b d x^{13} (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{17} b^2 d^2 x^{17} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 87, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{17}}{17}}+{\frac{ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){x}^{13}}{13}}+{\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}^{5}}{5}}+{a}^{2}{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94481, size = 111, normalized size = 1.35 \begin{align*} \frac{1}{17} \, b^{2} d^{2} x^{17} + \frac{2}{13} \,{\left (b^{2} c d + a b d^{2}\right )} x^{13} + \frac{1}{9} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{9} + \frac{2}{5} \,{\left (a b c^{2} + a^{2} c d\right )} x^{5} + a^{2} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.08849, size = 217, normalized size = 2.65 \begin{align*} \frac{1}{17} x^{17} d^{2} b^{2} + \frac{2}{13} x^{13} d c b^{2} + \frac{2}{13} x^{13} 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}{5} x^{5} c^{2} b a + \frac{2}{5} x^{5} d c a^{2} + x c^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.076386, size = 97, normalized size = 1.18 \begin{align*} a^{2} c^{2} x + \frac{b^{2} d^{2} x^{17}}{17} + x^{13} \left (\frac{2 a b d^{2}}{13} + \frac{2 b^{2} c d}{13}\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^{5} \left (\frac{2 a^{2} c d}{5} + \frac{2 a b c^{2}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09573, size = 123, normalized size = 1.5 \begin{align*} \frac{1}{17} \, b^{2} d^{2} x^{17} + \frac{2}{13} \, b^{2} c d x^{13} + \frac{2}{13} \, a b d^{2} x^{13} + \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}{5} \, a b c^{2} x^{5} + \frac{2}{5} \, a^{2} c d x^{5} + a^{2} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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