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

Optimal. Leaf size=154 \[ \frac{1}{17} d^2 x^{17} \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac{4}{13} c d x^{13} \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{1}{9} c^2 x^9 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a^2 c^4 x+\frac{2}{5} a c^3 x^5 (2 a d+b c)+\frac{2}{21} b d^3 x^{21} (a d+2 b c)+\frac{1}{25} b^2 d^4 x^{25} \]

[Out]

a^2*c^4*x + (2*a*c^3*(b*c + 2*a*d)*x^5)/5 + (c^2*(b^2*c^2 + 8*a*b*c*d + 6*a^2*d^
2)*x^9)/9 + (4*c*d*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^13)/13 + (d^2*(6*b^2*c^2 +
8*a*b*c*d + a^2*d^2)*x^17)/17 + (2*b*d^3*(2*b*c + a*d)*x^21)/21 + (b^2*d^4*x^25)
/25

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Rubi [A]  time = 0.235883, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{1}{17} d^2 x^{17} \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac{4}{13} c d x^{13} \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{1}{9} c^2 x^9 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a^2 c^4 x+\frac{2}{5} a c^3 x^5 (2 a d+b c)+\frac{2}{21} b d^3 x^{21} (a d+2 b c)+\frac{1}{25} b^2 d^4 x^{25} \]

Antiderivative was successfully verified.

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

[Out]

a^2*c^4*x + (2*a*c^3*(b*c + 2*a*d)*x^5)/5 + (c^2*(b^2*c^2 + 8*a*b*c*d + 6*a^2*d^
2)*x^9)/9 + (4*c*d*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^13)/13 + (d^2*(6*b^2*c^2 +
8*a*b*c*d + a^2*d^2)*x^17)/17 + (2*b*d^3*(2*b*c + a*d)*x^21)/21 + (b^2*d^4*x^25)
/25

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 a c^{3} x^{5} \left (2 a d + b c\right )}{5} + \frac{b^{2} d^{4} x^{25}}{25} + \frac{2 b d^{3} x^{21} \left (a d + 2 b c\right )}{21} + c^{4} \int a^{2}\, dx + \frac{c^{2} x^{9} \left (6 a^{2} d^{2} + 8 a b c d + b^{2} c^{2}\right )}{9} + \frac{4 c d x^{13} \left (a^{2} d^{2} + 3 a b c d + b^{2} c^{2}\right )}{13} + \frac{d^{2} x^{17} \left (a^{2} d^{2} + 8 a b c d + 6 b^{2} c^{2}\right )}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*c**3*x**5*(2*a*d + b*c)/5 + b**2*d**4*x**25/25 + 2*b*d**3*x**21*(a*d + 2*b*c
)/21 + c**4*Integral(a**2, x) + c**2*x**9*(6*a**2*d**2 + 8*a*b*c*d + b**2*c**2)/
9 + 4*c*d*x**13*(a**2*d**2 + 3*a*b*c*d + b**2*c**2)/13 + d**2*x**17*(a**2*d**2 +
8*a*b*c*d + 6*b**2*c**2)/17

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Mathematica [A]  time = 0.0624063, size = 154, normalized size = 1. \[ \frac{1}{17} d^2 x^{17} \left (a^2 d^2+8 a b c d+6 b^2 c^2\right )+\frac{4}{13} c d x^{13} \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{1}{9} c^2 x^9 \left (6 a^2 d^2+8 a b c d+b^2 c^2\right )+a^2 c^4 x+\frac{2}{5} a c^3 x^5 (2 a d+b c)+\frac{2}{21} b d^3 x^{21} (a d+2 b c)+\frac{1}{25} b^2 d^4 x^{25} \]

Antiderivative was successfully verified.

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

[Out]

a^2*c^4*x + (2*a*c^3*(b*c + 2*a*d)*x^5)/5 + (c^2*(b^2*c^2 + 8*a*b*c*d + 6*a^2*d^
2)*x^9)/9 + (4*c*d*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^13)/13 + (d^2*(6*b^2*c^2 +
8*a*b*c*d + a^2*d^2)*x^17)/17 + (2*b*d^3*(2*b*c + a*d)*x^21)/21 + (b^2*d^4*x^25)
/25

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Maple [A]  time = 0.001, size = 163, normalized size = 1.1 \[{\frac{{b}^{2}{d}^{4}{x}^{25}}{25}}+{\frac{ \left ( 2\,ab{d}^{4}+4\,{b}^{2}c{d}^{3} \right ){x}^{21}}{21}}+{\frac{ \left ({a}^{2}{d}^{4}+8\,abc{d}^{3}+6\,{b}^{2}{c}^{2}{d}^{2} \right ){x}^{17}}{17}}+{\frac{ \left ( 4\,{a}^{2}c{d}^{3}+12\,ab{c}^{2}{d}^{2}+4\,{b}^{2}{c}^{3}d \right ){x}^{13}}{13}}+{\frac{ \left ( 6\,{a}^{2}{c}^{2}{d}^{2}+8\,ab{c}^{3}d+{b}^{2}{c}^{4} \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,{a}^{2}{c}^{3}d+2\,ab{c}^{4} \right ){x}^{5}}{5}}+{a}^{2}{c}^{4}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^2*(d*x^4+c)^4,x)

[Out]

1/25*b^2*d^4*x^25+1/21*(2*a*b*d^4+4*b^2*c*d^3)*x^21+1/17*(a^2*d^4+8*a*b*c*d^3+6*
b^2*c^2*d^2)*x^17+1/13*(4*a^2*c*d^3+12*a*b*c^2*d^2+4*b^2*c^3*d)*x^13+1/9*(6*a^2*
c^2*d^2+8*a*b*c^3*d+b^2*c^4)*x^9+1/5*(4*a^2*c^3*d+2*a*b*c^4)*x^5+a^2*c^4*x

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Maxima [A]  time = 1.37164, size = 213, normalized size = 1.38 \[ \frac{1}{25} \, b^{2} d^{4} x^{25} + \frac{2}{21} \,{\left (2 \, b^{2} c d^{3} + a b d^{4}\right )} x^{21} + \frac{1}{17} \,{\left (6 \, b^{2} c^{2} d^{2} + 8 \, a b c d^{3} + a^{2} d^{4}\right )} x^{17} + \frac{4}{13} \,{\left (b^{2} c^{3} d + 3 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x^{13} + \frac{1}{9} \,{\left (b^{2} c^{4} + 8 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} x^{9} + a^{2} c^{4} x + \frac{2}{5} \,{\left (a b c^{4} + 2 \, a^{2} c^{3} d\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25*b^2*d^4*x^25 + 2/21*(2*b^2*c*d^3 + a*b*d^4)*x^21 + 1/17*(6*b^2*c^2*d^2 + 8*
a*b*c*d^3 + a^2*d^4)*x^17 + 4/13*(b^2*c^3*d + 3*a*b*c^2*d^2 + a^2*c*d^3)*x^13 +
1/9*(b^2*c^4 + 8*a*b*c^3*d + 6*a^2*c^2*d^2)*x^9 + a^2*c^4*x + 2/5*(a*b*c^4 + 2*a
^2*c^3*d)*x^5

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Fricas [A]  time = 0.188992, size = 1, normalized size = 0.01 \[ \frac{1}{25} x^{25} d^{4} b^{2} + \frac{4}{21} x^{21} d^{3} c b^{2} + \frac{2}{21} x^{21} d^{4} b a + \frac{6}{17} x^{17} d^{2} c^{2} b^{2} + \frac{8}{17} x^{17} d^{3} c b a + \frac{1}{17} x^{17} d^{4} a^{2} + \frac{4}{13} x^{13} d c^{3} b^{2} + \frac{12}{13} x^{13} d^{2} c^{2} b a + \frac{4}{13} x^{13} d^{3} c a^{2} + \frac{1}{9} x^{9} c^{4} b^{2} + \frac{8}{9} x^{9} d c^{3} b a + \frac{2}{3} x^{9} d^{2} c^{2} a^{2} + \frac{2}{5} x^{5} c^{4} b a + \frac{4}{5} x^{5} d c^{3} a^{2} + x c^{4} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25*x^25*d^4*b^2 + 4/21*x^21*d^3*c*b^2 + 2/21*x^21*d^4*b*a + 6/17*x^17*d^2*c^2*
b^2 + 8/17*x^17*d^3*c*b*a + 1/17*x^17*d^4*a^2 + 4/13*x^13*d*c^3*b^2 + 12/13*x^13
*d^2*c^2*b*a + 4/13*x^13*d^3*c*a^2 + 1/9*x^9*c^4*b^2 + 8/9*x^9*d*c^3*b*a + 2/3*x
^9*d^2*c^2*a^2 + 2/5*x^5*c^4*b*a + 4/5*x^5*d*c^3*a^2 + x*c^4*a^2

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Sympy [A]  time = 0.182983, size = 185, normalized size = 1.2 \[ a^{2} c^{4} x + \frac{b^{2} d^{4} x^{25}}{25} + x^{21} \left (\frac{2 a b d^{4}}{21} + \frac{4 b^{2} c d^{3}}{21}\right ) + x^{17} \left (\frac{a^{2} d^{4}}{17} + \frac{8 a b c d^{3}}{17} + \frac{6 b^{2} c^{2} d^{2}}{17}\right ) + x^{13} \left (\frac{4 a^{2} c d^{3}}{13} + \frac{12 a b c^{2} d^{2}}{13} + \frac{4 b^{2} c^{3} d}{13}\right ) + x^{9} \left (\frac{2 a^{2} c^{2} d^{2}}{3} + \frac{8 a b c^{3} d}{9} + \frac{b^{2} c^{4}}{9}\right ) + x^{5} \left (\frac{4 a^{2} c^{3} d}{5} + \frac{2 a b c^{4}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c**4*x + b**2*d**4*x**25/25 + x**21*(2*a*b*d**4/21 + 4*b**2*c*d**3/21) + x*
*17*(a**2*d**4/17 + 8*a*b*c*d**3/17 + 6*b**2*c**2*d**2/17) + x**13*(4*a**2*c*d**
3/13 + 12*a*b*c**2*d**2/13 + 4*b**2*c**3*d/13) + x**9*(2*a**2*c**2*d**2/3 + 8*a*
b*c**3*d/9 + b**2*c**4/9) + x**5*(4*a**2*c**3*d/5 + 2*a*b*c**4/5)

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GIAC/XCAS [A]  time = 0.209819, size = 234, normalized size = 1.52 \[ \frac{1}{25} \, b^{2} d^{4} x^{25} + \frac{4}{21} \, b^{2} c d^{3} x^{21} + \frac{2}{21} \, a b d^{4} x^{21} + \frac{6}{17} \, b^{2} c^{2} d^{2} x^{17} + \frac{8}{17} \, a b c d^{3} x^{17} + \frac{1}{17} \, a^{2} d^{4} x^{17} + \frac{4}{13} \, b^{2} c^{3} d x^{13} + \frac{12}{13} \, a b c^{2} d^{2} x^{13} + \frac{4}{13} \, a^{2} c d^{3} x^{13} + \frac{1}{9} \, b^{2} c^{4} x^{9} + \frac{8}{9} \, a b c^{3} d x^{9} + \frac{2}{3} \, a^{2} c^{2} d^{2} x^{9} + \frac{2}{5} \, a b c^{4} x^{5} + \frac{4}{5} \, a^{2} c^{3} d x^{5} + a^{2} c^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25*b^2*d^4*x^25 + 4/21*b^2*c*d^3*x^21 + 2/21*a*b*d^4*x^21 + 6/17*b^2*c^2*d^2*x
^17 + 8/17*a*b*c*d^3*x^17 + 1/17*a^2*d^4*x^17 + 4/13*b^2*c^3*d*x^13 + 12/13*a*b*
c^2*d^2*x^13 + 4/13*a^2*c*d^3*x^13 + 1/9*b^2*c^4*x^9 + 8/9*a*b*c^3*d*x^9 + 2/3*a
^2*c^2*d^2*x^9 + 2/5*a*b*c^4*x^5 + 4/5*a^2*c^3*d*x^5 + a^2*c^4*x