Optimal. Leaf size=50 \[ \frac{1}{9} d x^9 (a d+2 b c)+\frac{1}{5} c x^5 (2 a d+b c)+a c^2 x+\frac{1}{13} b d^2 x^{13} \]
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Rubi [A] time = 0.029443, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {373} \[ \frac{1}{9} d x^9 (a d+2 b c)+\frac{1}{5} c x^5 (2 a d+b c)+a c^2 x+\frac{1}{13} b d^2 x^{13} \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin{align*} \int \left (a+b x^4\right ) \left (c+d x^4\right )^2 \, dx &=\int \left (a c^2+c (b c+2 a d) x^4+d (2 b c+a d) x^8+b d^2 x^{12}\right ) \, dx\\ &=a c^2 x+\frac{1}{5} c (b c+2 a d) x^5+\frac{1}{9} d (2 b c+a d) x^9+\frac{1}{13} b d^2 x^{13}\\ \end{align*}
Mathematica [A] time = 0.011794, size = 50, normalized size = 1. \[ \frac{1}{9} d x^9 (a d+2 b c)+\frac{1}{5} c x^5 (2 a d+b c)+a c^2 x+\frac{1}{13} b d^2 x^{13} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 49, normalized size = 1. \begin{align*}{\frac{b{d}^{2}{x}^{13}}{13}}+{\frac{ \left ( a{d}^{2}+2\,bcd \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,acd+b{c}^{2} \right ){x}^{5}}{5}}+a{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948208, size = 65, normalized size = 1.3 \begin{align*} \frac{1}{13} \, b d^{2} x^{13} + \frac{1}{9} \,{\left (2 \, b c d + a d^{2}\right )} x^{9} + \frac{1}{5} \,{\left (b c^{2} + 2 \, a c d\right )} x^{5} + a c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.04988, size = 123, normalized size = 2.46 \begin{align*} \frac{1}{13} x^{13} d^{2} b + \frac{2}{9} x^{9} d c b + \frac{1}{9} x^{9} d^{2} a + \frac{1}{5} x^{5} c^{2} b + \frac{2}{5} x^{5} d c a + x c^{2} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.0685, size = 53, normalized size = 1.06 \begin{align*} a c^{2} x + \frac{b d^{2} x^{13}}{13} + x^{9} \left (\frac{a d^{2}}{9} + \frac{2 b c d}{9}\right ) + x^{5} \left (\frac{2 a c d}{5} + \frac{b c^{2}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09182, size = 68, normalized size = 1.36 \begin{align*} \frac{1}{13} \, b d^{2} x^{13} + \frac{2}{9} \, b c d x^{9} + \frac{1}{9} \, a d^{2} x^{9} + \frac{1}{5} \, b c^{2} x^{5} + \frac{2}{5} \, a c d x^{5} + a c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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