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

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

[Out]

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

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Rubi [A]  time = 0.0734854, 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 \[ a^2 c x+\frac{1}{9} b x^9 (2 a d+b c)+\frac{1}{5} a x^5 (a d+2 b c)+\frac{1}{13} b^2 d x^{13} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^2*(c + d*x^4),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} \int c\, dx + \frac{a x^{5} \left (a d + 2 b c\right )}{5} + \frac{b^{2} d x^{13}}{13} + \frac{b x^{9} \left (2 a d + b c\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**2*(d*x**4+c),x)

[Out]

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

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Mathematica [A]  time = 0.0136399, size = 50, normalized size = 1. \[ a^2 c x+\frac{1}{9} b x^9 (2 a d+b c)+\frac{1}{5} a x^5 (a d+2 b c)+\frac{1}{13} b^2 d x^{13} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^2*(c + d*x^4),x]

[Out]

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

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Maple [A]  time = 0.002, size = 49, normalized size = 1. \[{\frac{{b}^{2}d{x}^{13}}{13}}+{\frac{ \left ( 2\,abd+{b}^{2}c \right ){x}^{9}}{9}}+{\frac{ \left ({a}^{2}d+2\,abc \right ){x}^{5}}{5}}+{a}^{2}cx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^2*(d*x^4+c),x)

[Out]

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

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Maxima [A]  time = 1.39505, size = 65, normalized size = 1.3 \[ \frac{1}{13} \, b^{2} d x^{13} + \frac{1}{9} \,{\left (b^{2} c + 2 \, a b d\right )} x^{9} + \frac{1}{5} \,{\left (2 \, a b c + a^{2} d\right )} x^{5} + a^{2} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2*(d*x^4 + c),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.192196, size = 1, normalized size = 0.02 \[ \frac{1}{13} x^{13} d b^{2} + \frac{1}{9} x^{9} c b^{2} + \frac{2}{9} x^{9} d b a + \frac{2}{5} x^{5} c b a + \frac{1}{5} x^{5} d a^{2} + x c a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2*(d*x^4 + c),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.107578, size = 53, normalized size = 1.06 \[ a^{2} c x + \frac{b^{2} d x^{13}}{13} + x^{9} \left (\frac{2 a b d}{9} + \frac{b^{2} c}{9}\right ) + x^{5} \left (\frac{a^{2} d}{5} + \frac{2 a b c}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**2*(d*x**4+c),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.214651, size = 68, normalized size = 1.36 \[ \frac{1}{13} \, b^{2} d x^{13} + \frac{1}{9} \, b^{2} c x^{9} + \frac{2}{9} \, a b d x^{9} + \frac{2}{5} \, a b c x^{5} + \frac{1}{5} \, a^{2} d x^{5} + a^{2} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^2*(d*x^4 + c),x, algorithm="giac")

[Out]

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