3.49 \(\int \left (a+b x^4\right ) \left (c+d x^4\right )^2 \, dx\)

Optimal. Leaf size=50 \[ \frac{1}{9} d x^9 (a d+2 b c)+\frac{1}{5} c x^5 (2 a d+b c)+a c^2 x+\frac{1}{13} b d^2 x^{13} \]

[Out]

a*c^2*x + (c*(b*c + 2*a*d)*x^5)/5 + (d*(2*b*c + a*d)*x^9)/9 + (b*d^2*x^13)/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]

a*c^2*x + (c*(b*c + 2*a*d)*x^5)/5 + (d*(2*b*c + a*d)*x^9)/9 + (b*d^2*x^13)/13

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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]

b*d**2*x**13/13 + c**2*Integral(a, x) + c*x**5*(2*a*d + b*c)/5 + d*x**9*(a*d + 2
*b*c)/9

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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]

a*c^2*x + (c*(b*c + 2*a*d)*x^5)/5 + (d*(2*b*c + a*d)*x^9)/9 + (b*d^2*x^13)/13

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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]

1/13*b*d^2*x^13+1/9*(a*d^2+2*b*c*d)*x^9+1/5*(2*a*c*d+b*c^2)*x^5+a*c^2*x

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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]

1/13*b*d^2*x^13 + 1/9*(2*b*c*d + a*d^2)*x^9 + 1/5*(b*c^2 + 2*a*c*d)*x^5 + a*c^2*
x

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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]

1/13*x^13*d^2*b + 2/9*x^9*d*c*b + 1/9*x^9*d^2*a + 1/5*x^5*c^2*b + 2/5*x^5*d*c*a
+ x*c^2*a

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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]

a*c**2*x + b*d**2*x**13/13 + x**9*(a*d**2/9 + 2*b*c*d/9) + x**5*(2*a*c*d/5 + b*c
**2/5)

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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]

1/13*b*d^2*x^13 + 2/9*b*c*d*x^9 + 1/9*a*d^2*x^9 + 1/5*b*c^2*x^5 + 2/5*a*c*d*x^5
+ a*c^2*x