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 = 0.11, antiderivative size = 154, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, size = 154, normalized size = 1.00 \[ \frac {1}{17} d^2 x^{17} \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac {4}{13} c d x^{13} \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac {1}{9} c^2 x^9 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a^2 c^4 x+\frac {2}{5} a c^3 x^5 (2 a d+b c)+\frac {2}{21} b d^3 x^{21} (a d+2 b c)+\frac {1}{25} b^2 d^4 x^{25} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 173, normalized size = 1.12 \[ \frac {1}{25} x^{25} d^{4} b^{2} + \frac {4}{21} x^{21} d^{3} c b^{2} + \frac {2}{21} x^{21} d^{4} b a + \frac {6}{17} x^{17} d^{2} c^{2} b^{2} + \frac {8}{17} x^{17} d^{3} c b a + \frac {1}{17} x^{17} d^{4} a^{2} + \frac {4}{13} x^{13} d c^{3} b^{2} + \frac {12}{13} x^{13} d^{2} c^{2} b a + \frac {4}{13} x^{13} d^{3} c a^{2} + \frac {1}{9} x^{9} c^{4} b^{2} + \frac {8}{9} x^{9} d c^{3} b a + \frac {2}{3} x^{9} d^{2} c^{2} a^{2} + \frac {2}{5} x^{5} c^{4} b a + \frac {4}{5} x^{5} d c^{3} a^{2} + x c^{4} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 173, normalized size = 1.12 \[ \frac {1}{25} \, b^{2} d^{4} x^{25} + \frac {4}{21} \, b^{2} c d^{3} x^{21} + \frac {2}{21} \, a b d^{4} x^{21} + \frac {6}{17} \, b^{2} c^{2} d^{2} x^{17} + \frac {8}{17} \, a b c d^{3} x^{17} + \frac {1}{17} \, a^{2} d^{4} x^{17} + \frac {4}{13} \, b^{2} c^{3} d x^{13} + \frac {12}{13} \, a b c^{2} d^{2} x^{13} + \frac {4}{13} \, a^{2} c d^{3} x^{13} + \frac {1}{9} \, b^{2} c^{4} x^{9} + \frac {8}{9} \, a b c^{3} d x^{9} + \frac {2}{3} \, a^{2} c^{2} d^{2} x^{9} + \frac {2}{5} \, a b c^{4} x^{5} + \frac {4}{5} \, a^{2} c^{3} d x^{5} + a^{2} c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 163, normalized size = 1.06 \[ \frac {b^{2} d^{4} x^{25}}{25}+\frac {\left (2 a b \,d^{4}+4 b^{2} c \,d^{3}\right ) x^{21}}{21}+\frac {\left (a^{2} d^{4}+8 a b c \,d^{3}+6 b^{2} c^{2} d^{2}\right ) x^{17}}{17}+\frac {\left (4 a^{2} c \,d^{3}+12 a b \,c^{2} d^{2}+4 b^{2} c^{3} d \right ) x^{13}}{13}+\frac {\left (6 a^{2} c^{2} d^{2}+8 a b \,c^{3} d +b^{2} c^{4}\right ) x^{9}}{9}+a^{2} c^{4} x +\frac {\left (4 a^{2} c^{3} d +2 a b \,c^{4}\right ) x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 158, normalized size = 1.03 \[ \frac {1}{25} \, b^{2} d^{4} x^{25} + \frac {2}{21} \, {\left (2 \, b^{2} c d^{3} + a b d^{4}\right )} x^{21} + \frac {1}{17} \, {\left (6 \, b^{2} c^{2} d^{2} + 8 \, a b c d^{3} + a^{2} d^{4}\right )} x^{17} + \frac {4}{13} \, {\left (b^{2} c^{3} d + 3 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{13} + \frac {1}{9} \, {\left (b^{2} c^{4} + 8 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} x^{9} + a^{2} c^{4} x + \frac {2}{5} \, {\left (a b c^{4} + 2 \, a^{2} c^{3} d\right )} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 146, normalized size = 0.95 \[ x^9\,\left (\frac {2\,a^2\,c^2\,d^2}{3}+\frac {8\,a\,b\,c^3\,d}{9}+\frac {b^2\,c^4}{9}\right )+x^{17}\,\left (\frac {a^2\,d^4}{17}+\frac {8\,a\,b\,c\,d^3}{17}+\frac {6\,b^2\,c^2\,d^2}{17}\right )+a^2\,c^4\,x+\frac {b^2\,d^4\,x^{25}}{25}+\frac {2\,a\,c^3\,x^5\,\left (2\,a\,d+b\,c\right )}{5}+\frac {2\,b\,d^3\,x^{21}\,\left (a\,d+2\,b\,c\right )}{21}+\frac {4\,c\,d\,x^{13}\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 185, normalized size = 1.20 \[ a^{2} c^{4} x + \frac {b^{2} d^{4} x^{25}}{25} + x^{21} \left (\frac {2 a b d^{4}}{21} + \frac {4 b^{2} c d^{3}}{21}\right ) + x^{17} \left (\frac {a^{2} d^{4}}{17} + \frac {8 a b c d^{3}}{17} + \frac {6 b^{2} c^{2} d^{2}}{17}\right ) + x^{13} \left (\frac {4 a^{2} c d^{3}}{13} + \frac {12 a b c^{2} d^{2}}{13} + \frac {4 b^{2} c^{3} d}{13}\right ) + x^{9} \left (\frac {2 a^{2} c^{2} d^{2}}{3} + \frac {8 a b c^{3} d}{9} + \frac {b^{2} c^{4}}{9}\right ) + x^{5} \left (\frac {4 a^{2} c^{3} d}{5} + \frac {2 a b c^{4}}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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