Optimal. Leaf size=154 \[ \frac{1}{17} d^2 x^{17} \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac{4}{13} c d x^{13} \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{1}{9} c^2 x^9 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a^2 c^4 x+\frac{2}{5} a c^3 x^5 (2 a d+b c)+\frac{2}{21} b d^3 x^{21} (a d+2 b c)+\frac{1}{25} b^2 d^4 x^{25} \]
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Rubi [A] time = 0.113943, antiderivative size = 154, 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} \[ \frac{1}{17} d^2 x^{17} \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac{4}{13} c d x^{13} \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{1}{9} c^2 x^9 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a^2 c^4 x+\frac{2}{5} a c^3 x^5 (2 a d+b c)+\frac{2}{21} b d^3 x^{21} (a d+2 b c)+\frac{1}{25} b^2 d^4 x^{25} \]
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 )^4 \, dx &=\int \left (a^2 c^4+2 a c^3 (b c+2 a d) x^4+c^2 \left (b^2 c^2+8 a b c d+6 a^2 d^2\right ) x^8+4 c d \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^{12}+d^2 \left (6 b^2 c^2+8 a b c d+a^2 d^2\right ) x^{16}+2 b d^3 (2 b c+a d) x^{20}+b^2 d^4 x^{24}\right ) \, dx\\ &=a^2 c^4 x+\frac{2}{5} a c^3 (b c+2 a d) x^5+\frac{1}{9} c^2 \left (b^2 c^2+8 a b c d+6 a^2 d^2\right ) x^9+\frac{4}{13} c d \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^{13}+\frac{1}{17} d^2 \left (6 b^2 c^2+8 a b c d+a^2 d^2\right ) x^{17}+\frac{2}{21} b d^3 (2 b c+a d) x^{21}+\frac{1}{25} b^2 d^4 x^{25}\\ \end{align*}
Mathematica [A] time = 0.0314645, size = 154, normalized size = 1. \[ \frac{1}{17} d^2 x^{17} \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac{4}{13} c d x^{13} \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{1}{9} c^2 x^9 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a^2 c^4 x+\frac{2}{5} a c^3 x^5 (2 a d+b c)+\frac{2}{21} b d^3 x^{21} (a d+2 b c)+\frac{1}{25} b^2 d^4 x^{25} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 163, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{4}{x}^{25}}{25}}+{\frac{ \left ( 2\,ab{d}^{4}+4\,{b}^{2}c{d}^{3} \right ){x}^{21}}{21}}+{\frac{ \left ({a}^{2}{d}^{4}+8\,abc{d}^{3}+6\,{b}^{2}{c}^{2}{d}^{2} \right ){x}^{17}}{17}}+{\frac{ \left ( 4\,{a}^{2}c{d}^{3}+12\,ab{c}^{2}{d}^{2}+4\,{b}^{2}{c}^{3}d \right ){x}^{13}}{13}}+{\frac{ \left ( 6\,{a}^{2}{c}^{2}{d}^{2}+8\,ab{c}^{3}d+{b}^{2}{c}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,{a}^{2}{c}^{3}d+2\,ab{c}^{4} \right ){x}^{5}}{5}}+{a}^{2}{c}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949895, size = 213, normalized size = 1.38 \begin{align*} \frac{1}{25} \, b^{2} d^{4} x^{25} + \frac{2}{21} \,{\left (2 \, b^{2} c d^{3} + a b d^{4}\right )} x^{21} + \frac{1}{17} \,{\left (6 \, b^{2} c^{2} d^{2} + 8 \, a b c d^{3} + a^{2} d^{4}\right )} x^{17} + \frac{4}{13} \,{\left (b^{2} c^{3} d + 3 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{13} + \frac{1}{9} \,{\left (b^{2} c^{4} + 8 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} x^{9} + a^{2} c^{4} x + \frac{2}{5} \,{\left (a b c^{4} + 2 \, a^{2} c^{3} d\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11083, size = 413, normalized size = 2.68 \begin{align*} \frac{1}{25} x^{25} d^{4} b^{2} + \frac{4}{21} x^{21} d^{3} c b^{2} + \frac{2}{21} x^{21} d^{4} b a + \frac{6}{17} x^{17} d^{2} c^{2} b^{2} + \frac{8}{17} x^{17} d^{3} c b a + \frac{1}{17} x^{17} d^{4} a^{2} + \frac{4}{13} x^{13} d c^{3} b^{2} + \frac{12}{13} x^{13} d^{2} c^{2} b a + \frac{4}{13} x^{13} d^{3} c a^{2} + \frac{1}{9} x^{9} c^{4} b^{2} + \frac{8}{9} x^{9} d c^{3} b a + \frac{2}{3} x^{9} d^{2} c^{2} a^{2} + \frac{2}{5} x^{5} c^{4} b a + \frac{4}{5} x^{5} d c^{3} a^{2} + x c^{4} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.090721, size = 185, normalized size = 1.2 \begin{align*} a^{2} c^{4} x + \frac{b^{2} d^{4} x^{25}}{25} + x^{21} \left (\frac{2 a b d^{4}}{21} + \frac{4 b^{2} c d^{3}}{21}\right ) + x^{17} \left (\frac{a^{2} d^{4}}{17} + \frac{8 a b c d^{3}}{17} + \frac{6 b^{2} c^{2} d^{2}}{17}\right ) + x^{13} \left (\frac{4 a^{2} c d^{3}}{13} + \frac{12 a b c^{2} d^{2}}{13} + \frac{4 b^{2} c^{3} d}{13}\right ) + x^{9} \left (\frac{2 a^{2} c^{2} d^{2}}{3} + \frac{8 a b c^{3} d}{9} + \frac{b^{2} c^{4}}{9}\right ) + x^{5} \left (\frac{4 a^{2} c^{3} d}{5} + \frac{2 a b c^{4}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11187, size = 234, normalized size = 1.52 \begin{align*} \frac{1}{25} \, b^{2} d^{4} x^{25} + \frac{4}{21} \, b^{2} c d^{3} x^{21} + \frac{2}{21} \, a b d^{4} x^{21} + \frac{6}{17} \, b^{2} c^{2} d^{2} x^{17} + \frac{8}{17} \, a b c d^{3} x^{17} + \frac{1}{17} \, a^{2} d^{4} x^{17} + \frac{4}{13} \, b^{2} c^{3} d x^{13} + \frac{12}{13} \, a b c^{2} d^{2} x^{13} + \frac{4}{13} \, a^{2} c d^{3} x^{13} + \frac{1}{9} \, b^{2} c^{4} x^{9} + \frac{8}{9} \, a b c^{3} d x^{9} + \frac{2}{3} \, a^{2} c^{2} d^{2} x^{9} + \frac{2}{5} \, a b c^{4} x^{5} + \frac{4}{5} \, a^{2} c^{3} d x^{5} + a^{2} c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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