3.48 \(\int \left (a+b x^4\right ) \left (c+d x^4\right )^3 \, dx\)

Optimal. Leaf size=70 \[ \frac{1}{5} c^2 x^5 (3 a d+b c)+\frac{1}{13} d^2 x^{13} (a d+3 b c)+\frac{1}{3} c d x^9 (a d+b c)+a c^3 x+\frac{1}{17} b d^3 x^{17} \]

[Out]

a*c^3*x + (c^2*(b*c + 3*a*d)*x^5)/5 + (c*d*(b*c + a*d)*x^9)/3 + (d^2*(3*b*c + a*
d)*x^13)/13 + (b*d^3*x^17)/17

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Rubi [A]  time = 0.104301, antiderivative size = 70, 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 \[ \frac{1}{5} c^2 x^5 (3 a d+b c)+\frac{1}{13} d^2 x^{13} (a d+3 b c)+\frac{1}{3} c d x^9 (a d+b c)+a c^3 x+\frac{1}{17} b d^3 x^{17} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)*(c + d*x^4)^3,x]

[Out]

a*c^3*x + (c^2*(b*c + 3*a*d)*x^5)/5 + (c*d*(b*c + a*d)*x^9)/3 + (d^2*(3*b*c + a*
d)*x^13)/13 + (b*d^3*x^17)/17

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b d^{3} x^{17}}{17} + c^{3} \int a\, dx + \frac{c^{2} x^{5} \left (3 a d + b c\right )}{5} + \frac{c d x^{9} \left (a d + b c\right )}{3} + \frac{d^{2} x^{13} \left (a d + 3 b c\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)*(d*x**4+c)**3,x)

[Out]

b*d**3*x**17/17 + c**3*Integral(a, x) + c**2*x**5*(3*a*d + b*c)/5 + c*d*x**9*(a*
d + b*c)/3 + d**2*x**13*(a*d + 3*b*c)/13

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Mathematica [A]  time = 0.0311638, size = 70, normalized size = 1. \[ \frac{1}{5} c^2 x^5 (3 a d+b c)+\frac{1}{13} d^2 x^{13} (a d+3 b c)+\frac{1}{3} c d x^9 (a d+b c)+a c^3 x+\frac{1}{17} b d^3 x^{17} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)*(c + d*x^4)^3,x]

[Out]

a*c^3*x + (c^2*(b*c + 3*a*d)*x^5)/5 + (c*d*(b*c + a*d)*x^9)/3 + (d^2*(3*b*c + a*
d)*x^13)/13 + (b*d^3*x^17)/17

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Maple [A]  time = 0.001, size = 73, normalized size = 1. \[{\frac{b{d}^{3}{x}^{17}}{17}}+{\frac{ \left ( a{d}^{3}+3\,bc{d}^{2} \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,ac{d}^{2}+3\,b{c}^{2}d \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,a{c}^{2}d+b{c}^{3} \right ){x}^{5}}{5}}+a{c}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)*(d*x^4+c)^3,x)

[Out]

1/17*b*d^3*x^17+1/13*(a*d^3+3*b*c*d^2)*x^13+1/9*(3*a*c*d^2+3*b*c^2*d)*x^9+1/5*(3
*a*c^2*d+b*c^3)*x^5+a*c^3*x

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Maxima [A]  time = 1.3555, size = 95, normalized size = 1.36 \[ \frac{1}{17} \, b d^{3} x^{17} + \frac{1}{13} \,{\left (3 \, b c d^{2} + a d^{3}\right )} x^{13} + \frac{1}{3} \,{\left (b c^{2} d + a c d^{2}\right )} x^{9} + \frac{1}{5} \,{\left (b c^{3} + 3 \, a c^{2} d\right )} x^{5} + a c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)*(d*x^4 + c)^3,x, algorithm="maxima")

[Out]

1/17*b*d^3*x^17 + 1/13*(3*b*c*d^2 + a*d^3)*x^13 + 1/3*(b*c^2*d + a*c*d^2)*x^9 +
1/5*(b*c^3 + 3*a*c^2*d)*x^5 + a*c^3*x

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Fricas [A]  time = 0.191661, size = 1, normalized size = 0.01 \[ \frac{1}{17} x^{17} d^{3} b + \frac{3}{13} x^{13} d^{2} c b + \frac{1}{13} x^{13} d^{3} a + \frac{1}{3} x^{9} d c^{2} b + \frac{1}{3} x^{9} d^{2} c a + \frac{1}{5} x^{5} c^{3} b + \frac{3}{5} x^{5} d c^{2} a + x c^{3} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)*(d*x^4 + c)^3,x, algorithm="fricas")

[Out]

1/17*x^17*d^3*b + 3/13*x^13*d^2*c*b + 1/13*x^13*d^3*a + 1/3*x^9*d*c^2*b + 1/3*x^
9*d^2*c*a + 1/5*x^5*c^3*b + 3/5*x^5*d*c^2*a + x*c^3*a

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Sympy [A]  time = 0.122, size = 76, normalized size = 1.09 \[ a c^{3} x + \frac{b d^{3} x^{17}}{17} + x^{13} \left (\frac{a d^{3}}{13} + \frac{3 b c d^{2}}{13}\right ) + x^{9} \left (\frac{a c d^{2}}{3} + \frac{b c^{2} d}{3}\right ) + x^{5} \left (\frac{3 a c^{2} d}{5} + \frac{b c^{3}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)*(d*x**4+c)**3,x)

[Out]

a*c**3*x + b*d**3*x**17/17 + x**13*(a*d**3/13 + 3*b*c*d**2/13) + x**9*(a*c*d**2/
3 + b*c**2*d/3) + x**5*(3*a*c**2*d/5 + b*c**3/5)

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GIAC/XCAS [A]  time = 0.211977, size = 100, normalized size = 1.43 \[ \frac{1}{17} \, b d^{3} x^{17} + \frac{3}{13} \, b c d^{2} x^{13} + \frac{1}{13} \, a d^{3} x^{13} + \frac{1}{3} \, b c^{2} d x^{9} + \frac{1}{3} \, a c d^{2} x^{9} + \frac{1}{5} \, b c^{3} x^{5} + \frac{3}{5} \, a c^{2} d x^{5} + a c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)*(d*x^4 + c)^3,x, algorithm="giac")

[Out]

1/17*b*d^3*x^17 + 3/13*b*c*d^2*x^13 + 1/13*a*d^3*x^13 + 1/3*b*c^2*d*x^9 + 1/3*a*
c*d^2*x^9 + 1/5*b*c^3*x^5 + 3/5*a*c^2*d*x^5 + a*c^3*x