Optimal. Leaf size=82 \[ \frac{1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{13} b d x^{13} (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{17} b^2 d^2 x^{17} \]
[Out]
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Rubi [A] time = 0.118699, 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 \[ \frac{1}{9} x^9 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 x+\frac{2}{13} b d x^{13} (a d+b c)+\frac{2}{5} a c x^5 (a d+b c)+\frac{1}{17} b^2 d^2 x^{17} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)^2*(c + d*x^4)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 a c x^{5} \left (a d + b c\right )}{5} + \frac{b^{2} d^{2} x^{17}}{17} + \frac{2 b d x^{13} \left (a d + b c\right )}{13} + c^{2} \int a^{2}\, dx + x^{9} \left (\frac{a^{2} d^{2}}{9} + \frac{4 a b c d}{9} + \frac{b^{2} c^{2}}{9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**2*(d*x**4+c)**2,x)
[Out]
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Mathematica [A] time = 0.0316722, 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.
[In] Integrate[(a + b*x^4)^2*(c + d*x^4)^2,x]
[Out]
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Maple [A] time = 0.001, size = 87, normalized size = 1.1 \[{\frac{{b}^{2}{d}^{2}{x}^{17}}{17}}+{\frac{ \left ( 2\,ab{d}^{2}+2\,{b}^{2}cd \right ){x}^{13}}{13}}+{\frac{ \left ({a}^{2}{d}^{2}+4\,cabd+{b}^{2}{c}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,{a}^{2}cd+2\,ab{c}^{2} \right ){x}^{5}}{5}}+{a}^{2}{c}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^2*(d*x^4+c)^2,x)
[Out]
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Maxima [A] time = 1.37829, size = 111, normalized size = 1.35 \[ \frac{1}{17} \, b^{2} d^{2} x^{17} + \frac{2}{13} \,{\left (b^{2} c d + a b d^{2}\right )} x^{13} + \frac{1}{9} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{9} + \frac{2}{5} \,{\left (a b c^{2} + a^{2} c d\right )} x^{5} + a^{2} c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^2*(d*x^4 + c)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.190083, size = 1, normalized size = 0.01 \[ \frac{1}{17} x^{17} d^{2} b^{2} + \frac{2}{13} x^{13} d c b^{2} + \frac{2}{13} x^{13} d^{2} b a + \frac{1}{9} x^{9} c^{2} b^{2} + \frac{4}{9} x^{9} d c b a + \frac{1}{9} x^{9} d^{2} a^{2} + \frac{2}{5} x^{5} c^{2} b a + \frac{2}{5} x^{5} d c a^{2} + x c^{2} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^2*(d*x^4 + c)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.138062, size = 97, normalized size = 1.18 \[ a^{2} c^{2} x + \frac{b^{2} d^{2} x^{17}}{17} + x^{13} \left (\frac{2 a b d^{2}}{13} + \frac{2 b^{2} c d}{13}\right ) + x^{9} \left (\frac{a^{2} d^{2}}{9} + \frac{4 a b c d}{9} + \frac{b^{2} c^{2}}{9}\right ) + x^{5} \left (\frac{2 a^{2} c d}{5} + \frac{2 a b c^{2}}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)**2*(d*x**4+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.209447, size = 123, normalized size = 1.5 \[ \frac{1}{17} \, b^{2} d^{2} x^{17} + \frac{2}{13} \, b^{2} c d x^{13} + \frac{2}{13} \, a b d^{2} x^{13} + \frac{1}{9} \, b^{2} c^{2} x^{9} + \frac{4}{9} \, a b c d x^{9} + \frac{1}{9} \, a^{2} d^{2} x^{9} + \frac{2}{5} \, a b c^{2} x^{5} + \frac{2}{5} \, a^{2} c d x^{5} + a^{2} c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^2*(d*x^4 + c)^2,x, algorithm="giac")
[Out]