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 = 0.0740207, 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 \[ \frac{1}{9} d x^9 (a d+2 b c)+\frac{1}{5} c x^5 (2 a d+b c)+a c^2 x+\frac{1}{13} b d^2 x^{13} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)*(c + d*x^4)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b d^{2} x^{13}}{13} + c^{2} \int a\, dx + \frac{c x^{5} \left (2 a d + b c\right )}{5} + \frac{d x^{9} \left (a d + 2 b c\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)*(d*x**4+c)**2,x)
[Out]
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Mathematica [A] time = 0.0200953, 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.
[In] Integrate[(a + b*x^4)*(c + d*x^4)^2,x]
[Out]
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Maple [A] time = 0.001, size = 49, normalized size = 1. \[{\frac{b{d}^{2}{x}^{13}}{13}}+{\frac{ \left ( a{d}^{2}+2\,bcd \right ){x}^{9}}{9}}+{\frac{ \left ( 2\,acd+b{c}^{2} \right ){x}^{5}}{5}}+a{c}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)*(d*x^4+c)^2,x)
[Out]
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Maxima [A] time = 1.41285, size = 65, normalized size = 1.3 \[ \frac{1}{13} \, b d^{2} x^{13} + \frac{1}{9} \,{\left (2 \, b c d + a d^{2}\right )} x^{9} + \frac{1}{5} \,{\left (b c^{2} + 2 \, a c d\right )} x^{5} + a c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)*(d*x^4 + c)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.189588, size = 1, normalized size = 0.02 \[ \frac{1}{13} x^{13} d^{2} b + \frac{2}{9} x^{9} d c b + \frac{1}{9} x^{9} d^{2} a + \frac{1}{5} x^{5} c^{2} b + \frac{2}{5} x^{5} d c a + x c^{2} a \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)*(d*x^4 + c)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.104548, size = 53, normalized size = 1.06 \[ a c^{2} x + \frac{b d^{2} x^{13}}{13} + x^{9} \left (\frac{a d^{2}}{9} + \frac{2 b c d}{9}\right ) + x^{5} \left (\frac{2 a c d}{5} + \frac{b c^{2}}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)*(d*x**4+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.211371, size = 68, normalized size = 1.36 \[ \frac{1}{13} \, b d^{2} x^{13} + \frac{2}{9} \, b c d x^{9} + \frac{1}{9} \, a d^{2} x^{9} + \frac{1}{5} \, b c^{2} x^{5} + \frac{2}{5} \, a c d x^{5} + a c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)*(d*x^4 + c)^2,x, algorithm="giac")
[Out]