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

Optimal. Leaf size=94 \[ \frac{1}{5} c^3 x^5 (4 a d+b c)+\frac{2}{9} c^2 d x^9 (3 a d+2 b c)+\frac{1}{17} d^3 x^{17} (a d+4 b c)+\frac{2}{13} c d^2 x^{13} (2 a d+3 b c)+a c^4 x+\frac{1}{21} b d^4 x^{21} \]

[Out]

a*c^4*x + (c^3*(b*c + 4*a*d)*x^5)/5 + (2*c^2*d*(2*b*c + 3*a*d)*x^9)/9 + (2*c*d^2
*(3*b*c + 2*a*d)*x^13)/13 + (d^3*(4*b*c + a*d)*x^17)/17 + (b*d^4*x^21)/21

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Rubi [A]  time = 0.145413, antiderivative size = 94, 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 \[ \frac{1}{5} c^3 x^5 (4 a d+b c)+\frac{2}{9} c^2 d x^9 (3 a d+2 b c)+\frac{1}{17} d^3 x^{17} (a d+4 b c)+\frac{2}{13} c d^2 x^{13} (2 a d+3 b c)+a c^4 x+\frac{1}{21} b d^4 x^{21} \]

Antiderivative was successfully verified.

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

[Out]

a*c^4*x + (c^3*(b*c + 4*a*d)*x^5)/5 + (2*c^2*d*(2*b*c + 3*a*d)*x^9)/9 + (2*c*d^2
*(3*b*c + 2*a*d)*x^13)/13 + (d^3*(4*b*c + a*d)*x^17)/17 + (b*d^4*x^21)/21

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b d^{4} x^{21}}{21} + c^{4} \int a\, dx + \frac{c^{3} x^{5} \left (4 a d + b c\right )}{5} + \frac{2 c^{2} d x^{9} \left (3 a d + 2 b c\right )}{9} + \frac{2 c d^{2} x^{13} \left (2 a d + 3 b c\right )}{13} + \frac{d^{3} x^{17} \left (a d + 4 b c\right )}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*d**4*x**21/21 + c**4*Integral(a, x) + c**3*x**5*(4*a*d + b*c)/5 + 2*c**2*d*x**
9*(3*a*d + 2*b*c)/9 + 2*c*d**2*x**13*(2*a*d + 3*b*c)/13 + d**3*x**17*(a*d + 4*b*
c)/17

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Mathematica [A]  time = 0.0351537, size = 94, normalized size = 1. \[ \frac{1}{5} c^3 x^5 (4 a d+b c)+\frac{2}{9} c^2 d x^9 (3 a d+2 b c)+\frac{1}{17} d^3 x^{17} (a d+4 b c)+\frac{2}{13} c d^2 x^{13} (2 a d+3 b c)+a c^4 x+\frac{1}{21} b d^4 x^{21} \]

Antiderivative was successfully verified.

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

[Out]

a*c^4*x + (c^3*(b*c + 4*a*d)*x^5)/5 + (2*c^2*d*(2*b*c + 3*a*d)*x^9)/9 + (2*c*d^2
*(3*b*c + 2*a*d)*x^13)/13 + (d^3*(4*b*c + a*d)*x^17)/17 + (b*d^4*x^21)/21

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Maple [A]  time = 0.001, size = 97, normalized size = 1. \[{\frac{b{d}^{4}{x}^{21}}{21}}+{\frac{ \left ( a{d}^{4}+4\,bc{d}^{3} \right ){x}^{17}}{17}}+{\frac{ \left ( 4\,ac{d}^{3}+6\,{c}^{2}{d}^{2}b \right ){x}^{13}}{13}}+{\frac{ \left ( 6\,a{c}^{2}{d}^{2}+4\,{c}^{3}db \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,a{c}^{3}d+b{c}^{4} \right ){x}^{5}}{5}}+a{c}^{4}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)*(d*x^4+c)^4,x)

[Out]

1/21*b*d^4*x^21+1/17*(a*d^4+4*b*c*d^3)*x^17+1/13*(4*a*c*d^3+6*b*c^2*d^2)*x^13+1/
9*(6*a*c^2*d^2+4*b*c^3*d)*x^9+1/5*(4*a*c^3*d+b*c^4)*x^5+a*c^4*x

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Maxima [A]  time = 1.36215, size = 130, normalized size = 1.38 \[ \frac{1}{21} \, b d^{4} x^{21} + \frac{1}{17} \,{\left (4 \, b c d^{3} + a d^{4}\right )} x^{17} + \frac{2}{13} \,{\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{13} + \frac{2}{9} \,{\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} x^{9} + a c^{4} x + \frac{1}{5} \,{\left (b c^{4} + 4 \, a c^{3} d\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*b*d^4*x^21 + 1/17*(4*b*c*d^3 + a*d^4)*x^17 + 2/13*(3*b*c^2*d^2 + 2*a*c*d^3)
*x^13 + 2/9*(2*b*c^3*d + 3*a*c^2*d^2)*x^9 + a*c^4*x + 1/5*(b*c^4 + 4*a*c^3*d)*x^
5

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Fricas [A]  time = 0.191658, size = 1, normalized size = 0.01 \[ \frac{1}{21} x^{21} d^{4} b + \frac{4}{17} x^{17} d^{3} c b + \frac{1}{17} x^{17} d^{4} a + \frac{6}{13} x^{13} d^{2} c^{2} b + \frac{4}{13} x^{13} d^{3} c a + \frac{4}{9} x^{9} d c^{3} b + \frac{2}{3} x^{9} d^{2} c^{2} a + \frac{1}{5} x^{5} c^{4} b + \frac{4}{5} x^{5} d c^{3} a + x c^{4} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*x^21*d^4*b + 4/17*x^17*d^3*c*b + 1/17*x^17*d^4*a + 6/13*x^13*d^2*c^2*b + 4/
13*x^13*d^3*c*a + 4/9*x^9*d*c^3*b + 2/3*x^9*d^2*c^2*a + 1/5*x^5*c^4*b + 4/5*x^5*
d*c^3*a + x*c^4*a

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Sympy [A]  time = 0.149023, size = 107, normalized size = 1.14 \[ a c^{4} x + \frac{b d^{4} x^{21}}{21} + x^{17} \left (\frac{a d^{4}}{17} + \frac{4 b c d^{3}}{17}\right ) + x^{13} \left (\frac{4 a c d^{3}}{13} + \frac{6 b c^{2} d^{2}}{13}\right ) + x^{9} \left (\frac{2 a c^{2} d^{2}}{3} + \frac{4 b c^{3} d}{9}\right ) + x^{5} \left (\frac{4 a c^{3} d}{5} + \frac{b c^{4}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**4*x + b*d**4*x**21/21 + x**17*(a*d**4/17 + 4*b*c*d**3/17) + x**13*(4*a*c*d*
*3/13 + 6*b*c**2*d**2/13) + x**9*(2*a*c**2*d**2/3 + 4*b*c**3*d/9) + x**5*(4*a*c*
*3*d/5 + b*c**4/5)

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GIAC/XCAS [A]  time = 0.212669, size = 132, normalized size = 1.4 \[ \frac{1}{21} \, b d^{4} x^{21} + \frac{4}{17} \, b c d^{3} x^{17} + \frac{1}{17} \, a d^{4} x^{17} + \frac{6}{13} \, b c^{2} d^{2} x^{13} + \frac{4}{13} \, a c d^{3} x^{13} + \frac{4}{9} \, b c^{3} d x^{9} + \frac{2}{3} \, a c^{2} d^{2} x^{9} + \frac{1}{5} \, b c^{4} x^{5} + \frac{4}{5} \, a c^{3} d x^{5} + a c^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*b*d^4*x^21 + 4/17*b*c*d^3*x^17 + 1/17*a*d^4*x^17 + 6/13*b*c^2*d^2*x^13 + 4/
13*a*c*d^3*x^13 + 4/9*b*c^3*d*x^9 + 2/3*a*c^2*d^2*x^9 + 1/5*b*c^4*x^5 + 4/5*a*c^
3*d*x^5 + a*c^4*x