Optimal. Leaf size=82 \[ \frac{1}{7} x^7 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{1}{5} b d x^{10} (a d+b c)+\frac{1}{2} a c x^4 (a d+b c)+\frac{1}{13} b^2 d^2 x^{13} \]
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Rubi [A] time = 0.0460077, 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} \[ \frac{1}{7} x^7 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{1}{5} b d x^{10} (a d+b c)+\frac{1}{2} a c x^4 (a d+b c)+\frac{1}{13} b^2 d^2 x^{13} \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin{align*} \int \left (a+b x^3\right )^2 \left (c+d x^3\right )^2 \, dx &=\int \left (a^2 c^2+2 a c (b c+a d) x^3+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^6+2 b d (b c+a d) x^9+b^2 d^2 x^{12}\right ) \, dx\\ &=a^2 c^2 x+\frac{1}{2} a c (b c+a d) x^4+\frac{1}{7} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^7+\frac{1}{5} b d (b c+a d) x^{10}+\frac{1}{13} b^2 d^2 x^{13}\\ \end{align*}
Mathematica [A] time = 0.0110951, size = 82, normalized size = 1. \[ \frac{1}{7} x^7 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{1}{5} b d x^{10} (a d+b c)+\frac{1}{2} a c x^4 (a d+b c)+\frac{1}{13} b^2 d^2 x^{13} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 87, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{13}}{13}}+{\frac{ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){x}^{10}}{10}}+{\frac{ \left ({a}^{2}{d}^{2}+4\,cabd+{b}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){x}^{4}}{4}}+{a}^{2}{c}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967997, size = 111, normalized size = 1.35 \begin{align*} \frac{1}{13} \, b^{2} d^{2} x^{13} + \frac{1}{5} \,{\left (b^{2} c d + a b d^{2}\right )} x^{10} + \frac{1}{7} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{7} + a^{2} c^{2} x + \frac{1}{2} \,{\left (a b c^{2} + a^{2} c d\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30395, size = 215, normalized size = 2.62 \begin{align*} \frac{1}{13} x^{13} d^{2} b^{2} + \frac{1}{5} x^{10} d c b^{2} + \frac{1}{5} x^{10} d^{2} b a + \frac{1}{7} x^{7} c^{2} b^{2} + \frac{4}{7} x^{7} d c b a + \frac{1}{7} x^{7} d^{2} a^{2} + \frac{1}{2} x^{4} c^{2} b a + \frac{1}{2} x^{4} 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 = 0.112381, size = 90, normalized size = 1.1 \begin{align*} a^{2} c^{2} x + \frac{b^{2} d^{2} x^{13}}{13} + x^{10} \left (\frac{a b d^{2}}{5} + \frac{b^{2} c d}{5}\right ) + x^{7} \left (\frac{a^{2} d^{2}}{7} + \frac{4 a b c d}{7} + \frac{b^{2} c^{2}}{7}\right ) + x^{4} \left (\frac{a^{2} c d}{2} + \frac{a b c^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10344, size = 123, normalized size = 1.5 \begin{align*} \frac{1}{13} \, b^{2} d^{2} x^{13} + \frac{1}{5} \, b^{2} c d x^{10} + \frac{1}{5} \, a b d^{2} x^{10} + \frac{1}{7} \, b^{2} c^{2} x^{7} + \frac{4}{7} \, a b c d x^{7} + \frac{1}{7} \, a^{2} d^{2} x^{7} + \frac{1}{2} \, a b c^{2} x^{4} + \frac{1}{2} \, a^{2} c d x^{4} + a^{2} c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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