3.60 \(\int \left (a+b x^2+c x^4\right )^3 \left (a d+a e x+(b d+a f) x^2+b e x^3+(c d+b f) x^4+c e x^5+c f x^6\right ) \, dx\)

Optimal. Leaf size=416 \[ a^4 d x+\frac{1}{2} a^4 e x^2+\frac{1}{3} a^3 x^3 (a f+4 b d)+a^3 b e x^4+\frac{2}{5} a^2 x^5 \left (2 a b f+2 a c d+3 b^2 d\right )+\frac{1}{3} a^2 e x^6 \left (2 a c+3 b^2\right )+\frac{1}{10} e x^{10} \left (6 a^2 c^2+12 a b^2 c+b^4\right )+\frac{2}{7} a x^7 \left (2 a^2 c f+3 a b^2 f+6 a b c d+2 b^3 d\right )+\frac{1}{11} x^{11} \left (6 a^2 c^2 f+12 a b^2 c f+12 a b c^2 d+b^4 f+4 b^3 c d\right )+\frac{1}{9} x^9 \left (12 a^2 b c f+6 a^2 c^2 d+4 a b^3 f+12 a b^2 c d+b^4 d\right )+\frac{2}{15} c^2 x^{15} \left (2 a c f+3 b^2 f+2 b c d\right )+\frac{1}{7} c^2 e x^{14} \left (2 a c+3 b^2\right )+\frac{1}{3} b c e x^{12} \left (3 a c+b^2\right )+\frac{1}{2} a b e x^8 \left (3 a c+b^2\right )+\frac{2}{13} c x^{13} \left (6 a b c f+2 a c^2 d+2 b^3 f+3 b^2 c d\right )+\frac{1}{17} c^3 x^{17} (4 b f+c d)+\frac{1}{4} b c^3 e x^{16}+\frac{1}{18} c^4 e x^{18}+\frac{1}{19} c^4 f x^{19} \]

[Out]

a^4*d*x + (a^4*e*x^2)/2 + (a^3*(4*b*d + a*f)*x^3)/3 + a^3*b*e*x^4 + (2*a^2*(3*b^
2*d + 2*a*c*d + 2*a*b*f)*x^5)/5 + (a^2*(3*b^2 + 2*a*c)*e*x^6)/3 + (2*a*(2*b^3*d
+ 6*a*b*c*d + 3*a*b^2*f + 2*a^2*c*f)*x^7)/7 + (a*b*(b^2 + 3*a*c)*e*x^8)/2 + ((b^
4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 4*a*b^3*f + 12*a^2*b*c*f)*x^9)/9 + ((b^4 + 12
*a*b^2*c + 6*a^2*c^2)*e*x^10)/10 + ((4*b^3*c*d + 12*a*b*c^2*d + b^4*f + 12*a*b^2
*c*f + 6*a^2*c^2*f)*x^11)/11 + (b*c*(b^2 + 3*a*c)*e*x^12)/3 + (2*c*(3*b^2*c*d +
2*a*c^2*d + 2*b^3*f + 6*a*b*c*f)*x^13)/13 + (c^2*(3*b^2 + 2*a*c)*e*x^14)/7 + (2*
c^2*(2*b*c*d + 3*b^2*f + 2*a*c*f)*x^15)/15 + (b*c^3*e*x^16)/4 + (c^3*(c*d + 4*b*
f)*x^17)/17 + (c^4*e*x^18)/18 + (c^4*f*x^19)/19

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Rubi [A]  time = 1.22087, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 63, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.016 \[ a^4 d x+\frac{1}{2} a^4 e x^2+\frac{1}{3} a^3 x^3 (a f+4 b d)+a^3 b e x^4+\frac{2}{5} a^2 x^5 \left (2 a b f+2 a c d+3 b^2 d\right )+\frac{1}{3} a^2 e x^6 \left (2 a c+3 b^2\right )+\frac{1}{10} e x^{10} \left (6 a^2 c^2+12 a b^2 c+b^4\right )+\frac{2}{7} a x^7 \left (2 a^2 c f+3 a b^2 f+6 a b c d+2 b^3 d\right )+\frac{1}{11} x^{11} \left (6 a^2 c^2 f+12 a b^2 c f+12 a b c^2 d+b^4 f+4 b^3 c d\right )+\frac{1}{9} x^9 \left (12 a^2 b c f+6 a^2 c^2 d+4 a b^3 f+12 a b^2 c d+b^4 d\right )+\frac{2}{15} c^2 x^{15} \left (2 a c f+3 b^2 f+2 b c d\right )+\frac{1}{7} c^2 e x^{14} \left (2 a c+3 b^2\right )+\frac{1}{3} b c e x^{12} \left (3 a c+b^2\right )+\frac{1}{2} a b e x^8 \left (3 a c+b^2\right )+\frac{2}{13} c x^{13} \left (6 a b c f+2 a c^2 d+2 b^3 f+3 b^2 c d\right )+\frac{1}{17} c^3 x^{17} (4 b f+c d)+\frac{1}{4} b c^3 e x^{16}+\frac{1}{18} c^4 e x^{18}+\frac{1}{19} c^4 f x^{19} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^3*(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6),x]

[Out]

a^4*d*x + (a^4*e*x^2)/2 + (a^3*(4*b*d + a*f)*x^3)/3 + a^3*b*e*x^4 + (2*a^2*(3*b^
2*d + 2*a*c*d + 2*a*b*f)*x^5)/5 + (a^2*(3*b^2 + 2*a*c)*e*x^6)/3 + (2*a*(2*b^3*d
+ 6*a*b*c*d + 3*a*b^2*f + 2*a^2*c*f)*x^7)/7 + (a*b*(b^2 + 3*a*c)*e*x^8)/2 + ((b^
4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 4*a*b^3*f + 12*a^2*b*c*f)*x^9)/9 + ((b^4 + 12
*a*b^2*c + 6*a^2*c^2)*e*x^10)/10 + ((4*b^3*c*d + 12*a*b*c^2*d + b^4*f + 12*a*b^2
*c*f + 6*a^2*c^2*f)*x^11)/11 + (b*c*(b^2 + 3*a*c)*e*x^12)/3 + (2*c*(3*b^2*c*d +
2*a*c^2*d + 2*b^3*f + 6*a*b*c*f)*x^13)/13 + (c^2*(3*b^2 + 2*a*c)*e*x^14)/7 + (2*
c^2*(2*b*c*d + 3*b^2*f + 2*a*c*f)*x^15)/15 + (b*c^3*e*x^16)/4 + (c^3*(c*d + 4*b*
f)*x^17)/17 + (c^4*e*x^18)/18 + (c^4*f*x^19)/19

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{4} e \int x\, dx + a^{4} \int d\, dx + a^{3} b e x^{4} + \frac{a^{3} x^{3} \left (a f + 4 b d\right )}{3} + \frac{a^{2} e x^{6} \left (2 a c + 3 b^{2}\right )}{3} + \frac{2 a^{2} x^{5} \left (2 a b f + 2 a c d + 3 b^{2} d\right )}{5} + \frac{a b e x^{8} \left (3 a c + b^{2}\right )}{2} + \frac{2 a x^{7} \left (2 a^{2} c f + 3 a b^{2} f + 6 a b c d + 2 b^{3} d\right )}{7} + \frac{b c^{3} e x^{16}}{4} + \frac{b c e x^{12} \left (3 a c + b^{2}\right )}{3} + \frac{c^{4} e x^{18}}{18} + \frac{c^{4} f x^{19}}{19} + \frac{c^{3} x^{17} \left (4 b f + c d\right )}{17} + \frac{c^{2} e x^{14} \left (2 a c + 3 b^{2}\right )}{7} + \frac{2 c^{2} x^{15} \left (2 a c f + 3 b^{2} f + 2 b c d\right )}{15} + \frac{2 c x^{13} \left (6 a b c f + 2 a c^{2} d + 2 b^{3} f + 3 b^{2} c d\right )}{13} + \frac{e x^{10} \left (6 a^{2} c^{2} + 12 a b^{2} c + b^{4}\right )}{10} + x^{11} \left (\frac{6 a^{2} c^{2} f}{11} + \frac{12 a b^{2} c f}{11} + \frac{12 a b c^{2} d}{11} + \frac{b^{4} f}{11} + \frac{4 b^{3} c d}{11}\right ) + x^{9} \left (\frac{4 a^{2} b c f}{3} + \frac{2 a^{2} c^{2} d}{3} + \frac{4 a b^{3} f}{9} + \frac{4 a b^{2} c d}{3} + \frac{b^{4} d}{9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**3*(a*d+a*e*x+(a*f+b*d)*x**2+b*e*x**3+(b*f+c*d)*x**4+c*e*x**5+c*f*x**6),x)

[Out]

a**4*e*Integral(x, x) + a**4*Integral(d, x) + a**3*b*e*x**4 + a**3*x**3*(a*f + 4
*b*d)/3 + a**2*e*x**6*(2*a*c + 3*b**2)/3 + 2*a**2*x**5*(2*a*b*f + 2*a*c*d + 3*b*
*2*d)/5 + a*b*e*x**8*(3*a*c + b**2)/2 + 2*a*x**7*(2*a**2*c*f + 3*a*b**2*f + 6*a*
b*c*d + 2*b**3*d)/7 + b*c**3*e*x**16/4 + b*c*e*x**12*(3*a*c + b**2)/3 + c**4*e*x
**18/18 + c**4*f*x**19/19 + c**3*x**17*(4*b*f + c*d)/17 + c**2*e*x**14*(2*a*c +
3*b**2)/7 + 2*c**2*x**15*(2*a*c*f + 3*b**2*f + 2*b*c*d)/15 + 2*c*x**13*(6*a*b*c*
f + 2*a*c**2*d + 2*b**3*f + 3*b**2*c*d)/13 + e*x**10*(6*a**2*c**2 + 12*a*b**2*c
+ b**4)/10 + x**11*(6*a**2*c**2*f/11 + 12*a*b**2*c*f/11 + 12*a*b*c**2*d/11 + b**
4*f/11 + 4*b**3*c*d/11) + x**9*(4*a**2*b*c*f/3 + 2*a**2*c**2*d/3 + 4*a*b**3*f/9
+ 4*a*b**2*c*d/3 + b**4*d/9)

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Mathematica [A]  time = 0.208483, size = 416, normalized size = 1. \[ a^4 d x+\frac{1}{2} a^4 e x^2+\frac{1}{3} a^3 x^3 (a f+4 b d)+a^3 b e x^4+\frac{2}{5} a^2 x^5 \left (2 a b f+2 a c d+3 b^2 d\right )+\frac{1}{3} a^2 e x^6 \left (2 a c+3 b^2\right )+\frac{1}{10} e x^{10} \left (6 a^2 c^2+12 a b^2 c+b^4\right )+\frac{2}{7} a x^7 \left (2 a^2 c f+3 a b^2 f+6 a b c d+2 b^3 d\right )+\frac{1}{11} x^{11} \left (6 a^2 c^2 f+12 a b^2 c f+12 a b c^2 d+b^4 f+4 b^3 c d\right )+\frac{1}{9} x^9 \left (12 a^2 b c f+6 a^2 c^2 d+4 a b^3 f+12 a b^2 c d+b^4 d\right )+\frac{2}{15} c^2 x^{15} \left (2 a c f+3 b^2 f+2 b c d\right )+\frac{1}{7} c^2 e x^{14} \left (2 a c+3 b^2\right )+\frac{1}{3} b c e x^{12} \left (3 a c+b^2\right )+\frac{1}{2} a b e x^8 \left (3 a c+b^2\right )+\frac{2}{13} c x^{13} \left (6 a b c f+2 a c^2 d+2 b^3 f+3 b^2 c d\right )+\frac{1}{17} c^3 x^{17} (4 b f+c d)+\frac{1}{4} b c^3 e x^{16}+\frac{1}{18} c^4 e x^{18}+\frac{1}{19} c^4 f x^{19} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^3*(a*d + a*e*x + (b*d + a*f)*x^2 + b*e*x^3 + (c*d + b*f)*x^4 + c*e*x^5 + c*f*x^6),x]

[Out]

a^4*d*x + (a^4*e*x^2)/2 + (a^3*(4*b*d + a*f)*x^3)/3 + a^3*b*e*x^4 + (2*a^2*(3*b^
2*d + 2*a*c*d + 2*a*b*f)*x^5)/5 + (a^2*(3*b^2 + 2*a*c)*e*x^6)/3 + (2*a*(2*b^3*d
+ 6*a*b*c*d + 3*a*b^2*f + 2*a^2*c*f)*x^7)/7 + (a*b*(b^2 + 3*a*c)*e*x^8)/2 + ((b^
4*d + 12*a*b^2*c*d + 6*a^2*c^2*d + 4*a*b^3*f + 12*a^2*b*c*f)*x^9)/9 + ((b^4 + 12
*a*b^2*c + 6*a^2*c^2)*e*x^10)/10 + ((4*b^3*c*d + 12*a*b*c^2*d + b^4*f + 12*a*b^2
*c*f + 6*a^2*c^2*f)*x^11)/11 + (b*c*(b^2 + 3*a*c)*e*x^12)/3 + (2*c*(3*b^2*c*d +
2*a*c^2*d + 2*b^3*f + 6*a*b*c*f)*x^13)/13 + (c^2*(3*b^2 + 2*a*c)*e*x^14)/7 + (2*
c^2*(2*b*c*d + 3*b^2*f + 2*a*c*f)*x^15)/15 + (b*c^3*e*x^16)/4 + (c^3*(c*d + 4*b*
f)*x^17)/17 + (c^4*e*x^18)/18 + (c^4*f*x^19)/19

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Maple [B]  time = 0.002, size = 829, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^3*(a*d+a*e*x+(a*f+b*d)*x^2+b*e*x^3+(b*f+c*d)*x^4+c*e*x^5+c*f*x^6),x)

[Out]

1/19*c^4*f*x^19+1/18*c^4*e*x^18+1/17*(3*b*c^3*f+c^3*(b*f+c*d))*x^17+1/4*b*c^3*e*
x^16+1/15*((a*c^2+2*b^2*c+c*(2*a*c+b^2))*c*f+3*c^2*b*(b*f+c*d)+c^3*(a*f+b*d))*x^
15+1/14*((a*c^2+2*b^2*c+c*(2*a*c+b^2))*c*e+3*c^2*b^2*e+a*c^3*e)*x^14+1/13*((4*a*
b*c+b*(2*a*c+b^2))*c*f+(a*c^2+2*b^2*c+c*(2*a*c+b^2))*(b*f+c*d)+3*c^2*b*(a*f+b*d)
+c^3*a*d)*x^13+1/12*((4*a*b*c+b*(2*a*c+b^2))*c*e+(a*c^2+2*b^2*c+c*(2*a*c+b^2))*b
*e+3*a*b*c^2*e)*x^12+1/11*((a*(2*a*c+b^2)+2*a*b^2+a^2*c)*c*f+(4*a*b*c+b*(2*a*c+b
^2))*(b*f+c*d)+(a*c^2+2*b^2*c+c*(2*a*c+b^2))*(a*f+b*d)+3*a*b*c^2*d)*x^11+1/10*((
a*(2*a*c+b^2)+2*a*b^2+a^2*c)*c*e+(4*a*b*c+b*(2*a*c+b^2))*b*e+(a*c^2+2*b^2*c+c*(2
*a*c+b^2))*a*e)*x^10+1/9*(3*a^2*b*c*f+(a*(2*a*c+b^2)+2*a*b^2+a^2*c)*(b*f+c*d)+(4
*a*b*c+b*(2*a*c+b^2))*(a*f+b*d)+(a*c^2+2*b^2*c+c*(2*a*c+b^2))*a*d)*x^9+1/8*(3*a^
2*b*c*e+(a*(2*a*c+b^2)+2*a*b^2+a^2*c)*b*e+(4*a*b*c+b*(2*a*c+b^2))*a*e)*x^8+1/7*(
a^3*c*f+3*a^2*b*(b*f+c*d)+(a*(2*a*c+b^2)+2*a*b^2+a^2*c)*(a*f+b*d)+(4*a*b*c+b*(2*
a*c+b^2))*a*d)*x^7+1/6*(a^3*c*e+3*a^2*b^2*e+(a*(2*a*c+b^2)+2*a*b^2+a^2*c)*a*e)*x
^6+1/5*(a^3*(b*f+c*d)+3*a^2*b*(a*f+b*d)+(a*(2*a*c+b^2)+2*a*b^2+a^2*c)*a*d)*x^5+a
^3*b*e*x^4+1/3*(a^3*(a*f+b*d)+3*a^3*b*d)*x^3+1/2*a^4*e*x^2+a^4*d*x

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Maxima [A]  time = 0.710354, size = 564, normalized size = 1.36 \[ \frac{1}{19} \, c^{4} f x^{19} + \frac{1}{18} \, c^{4} e x^{18} + \frac{1}{4} \, b c^{3} e x^{16} + \frac{1}{17} \,{\left (c^{4} d + 4 \, b c^{3} f\right )} x^{17} + \frac{1}{7} \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e x^{14} + \frac{2}{15} \,{\left (2 \, b c^{3} d +{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} f\right )} x^{15} + \frac{1}{3} \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e x^{12} + \frac{2}{13} \,{\left ({\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d + 2 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} f\right )} x^{13} + \frac{1}{10} \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e x^{10} + \frac{1}{11} \,{\left (4 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} f\right )} x^{11} + \frac{1}{2} \,{\left (a b^{3} + 3 \, a^{2} b c\right )} e x^{8} + \frac{1}{9} \,{\left ({\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d + 4 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} f\right )} x^{9} + a^{3} b e x^{4} + \frac{1}{3} \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e x^{6} + \frac{2}{7} \,{\left (2 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} f\right )} x^{7} + \frac{1}{2} \, a^{4} e x^{2} + a^{4} d x + \frac{2}{5} \,{\left (2 \, a^{3} b f +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d\right )} x^{5} + \frac{1}{3} \,{\left (4 \, a^{3} b d + a^{4} f\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*f*x^6 + c*e*x^5 + b*e*x^3 + (c*d + b*f)*x^4 + a*e*x + (b*d + a*f)*x^2 + a*d)*(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")

[Out]

1/19*c^4*f*x^19 + 1/18*c^4*e*x^18 + 1/4*b*c^3*e*x^16 + 1/17*(c^4*d + 4*b*c^3*f)*
x^17 + 1/7*(3*b^2*c^2 + 2*a*c^3)*e*x^14 + 2/15*(2*b*c^3*d + (3*b^2*c^2 + 2*a*c^3
)*f)*x^15 + 1/3*(b^3*c + 3*a*b*c^2)*e*x^12 + 2/13*((3*b^2*c^2 + 2*a*c^3)*d + 2*(
b^3*c + 3*a*b*c^2)*f)*x^13 + 1/10*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e*x^10 + 1/11*(
4*(b^3*c + 3*a*b*c^2)*d + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*f)*x^11 + 1/2*(a*b^3 +
3*a^2*b*c)*e*x^8 + 1/9*((b^4 + 12*a*b^2*c + 6*a^2*c^2)*d + 4*(a*b^3 + 3*a^2*b*c)
*f)*x^9 + a^3*b*e*x^4 + 1/3*(3*a^2*b^2 + 2*a^3*c)*e*x^6 + 2/7*(2*(a*b^3 + 3*a^2*
b*c)*d + (3*a^2*b^2 + 2*a^3*c)*f)*x^7 + 1/2*a^4*e*x^2 + a^4*d*x + 2/5*(2*a^3*b*f
+ (3*a^2*b^2 + 2*a^3*c)*d)*x^5 + 1/3*(4*a^3*b*d + a^4*f)*x^3

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Fricas [A]  time = 0.248385, size = 1, normalized size = 0. \[ \frac{1}{19} x^{19} f c^{4} + \frac{1}{18} x^{18} e c^{4} + \frac{1}{17} x^{17} d c^{4} + \frac{4}{17} x^{17} f c^{3} b + \frac{1}{4} x^{16} e c^{3} b + \frac{4}{15} x^{15} d c^{3} b + \frac{2}{5} x^{15} f c^{2} b^{2} + \frac{4}{15} x^{15} f c^{3} a + \frac{3}{7} x^{14} e c^{2} b^{2} + \frac{2}{7} x^{14} e c^{3} a + \frac{6}{13} x^{13} d c^{2} b^{2} + \frac{4}{13} x^{13} f c b^{3} + \frac{4}{13} x^{13} d c^{3} a + \frac{12}{13} x^{13} f c^{2} b a + \frac{1}{3} x^{12} e c b^{3} + x^{12} e c^{2} b a + \frac{4}{11} x^{11} d c b^{3} + \frac{1}{11} x^{11} f b^{4} + \frac{12}{11} x^{11} d c^{2} b a + \frac{12}{11} x^{11} f c b^{2} a + \frac{6}{11} x^{11} f c^{2} a^{2} + \frac{1}{10} x^{10} e b^{4} + \frac{6}{5} x^{10} e c b^{2} a + \frac{3}{5} x^{10} e c^{2} a^{2} + \frac{1}{9} x^{9} d b^{4} + \frac{4}{3} x^{9} d c b^{2} a + \frac{4}{9} x^{9} f b^{3} a + \frac{2}{3} x^{9} d c^{2} a^{2} + \frac{4}{3} x^{9} f c b a^{2} + \frac{1}{2} x^{8} e b^{3} a + \frac{3}{2} x^{8} e c b a^{2} + \frac{4}{7} x^{7} d b^{3} a + \frac{12}{7} x^{7} d c b a^{2} + \frac{6}{7} x^{7} f b^{2} a^{2} + \frac{4}{7} x^{7} f c a^{3} + x^{6} e b^{2} a^{2} + \frac{2}{3} x^{6} e c a^{3} + \frac{6}{5} x^{5} d b^{2} a^{2} + \frac{4}{5} x^{5} d c a^{3} + \frac{4}{5} x^{5} f b a^{3} + x^{4} e b a^{3} + \frac{4}{3} x^{3} d b a^{3} + \frac{1}{3} x^{3} f a^{4} + \frac{1}{2} x^{2} e a^{4} + x d a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*f*x^6 + c*e*x^5 + b*e*x^3 + (c*d + b*f)*x^4 + a*e*x + (b*d + a*f)*x^2 + a*d)*(c*x^4 + b*x^2 + a)^3,x, algorithm="fricas")

[Out]

1/19*x^19*f*c^4 + 1/18*x^18*e*c^4 + 1/17*x^17*d*c^4 + 4/17*x^17*f*c^3*b + 1/4*x^
16*e*c^3*b + 4/15*x^15*d*c^3*b + 2/5*x^15*f*c^2*b^2 + 4/15*x^15*f*c^3*a + 3/7*x^
14*e*c^2*b^2 + 2/7*x^14*e*c^3*a + 6/13*x^13*d*c^2*b^2 + 4/13*x^13*f*c*b^3 + 4/13
*x^13*d*c^3*a + 12/13*x^13*f*c^2*b*a + 1/3*x^12*e*c*b^3 + x^12*e*c^2*b*a + 4/11*
x^11*d*c*b^3 + 1/11*x^11*f*b^4 + 12/11*x^11*d*c^2*b*a + 12/11*x^11*f*c*b^2*a + 6
/11*x^11*f*c^2*a^2 + 1/10*x^10*e*b^4 + 6/5*x^10*e*c*b^2*a + 3/5*x^10*e*c^2*a^2 +
1/9*x^9*d*b^4 + 4/3*x^9*d*c*b^2*a + 4/9*x^9*f*b^3*a + 2/3*x^9*d*c^2*a^2 + 4/3*x
^9*f*c*b*a^2 + 1/2*x^8*e*b^3*a + 3/2*x^8*e*c*b*a^2 + 4/7*x^7*d*b^3*a + 12/7*x^7*
d*c*b*a^2 + 6/7*x^7*f*b^2*a^2 + 4/7*x^7*f*c*a^3 + x^6*e*b^2*a^2 + 2/3*x^6*e*c*a^
3 + 6/5*x^5*d*b^2*a^2 + 4/5*x^5*d*c*a^3 + 4/5*x^5*f*b*a^3 + x^4*e*b*a^3 + 4/3*x^
3*d*b*a^3 + 1/3*x^3*f*a^4 + 1/2*x^2*e*a^4 + x*d*a^4

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Sympy [A]  time = 0.359606, size = 503, normalized size = 1.21 \[ a^{4} d x + \frac{a^{4} e x^{2}}{2} + a^{3} b e x^{4} + \frac{b c^{3} e x^{16}}{4} + \frac{c^{4} e x^{18}}{18} + \frac{c^{4} f x^{19}}{19} + x^{17} \left (\frac{4 b c^{3} f}{17} + \frac{c^{4} d}{17}\right ) + x^{15} \left (\frac{4 a c^{3} f}{15} + \frac{2 b^{2} c^{2} f}{5} + \frac{4 b c^{3} d}{15}\right ) + x^{14} \left (\frac{2 a c^{3} e}{7} + \frac{3 b^{2} c^{2} e}{7}\right ) + x^{13} \left (\frac{12 a b c^{2} f}{13} + \frac{4 a c^{3} d}{13} + \frac{4 b^{3} c f}{13} + \frac{6 b^{2} c^{2} d}{13}\right ) + x^{12} \left (a b c^{2} e + \frac{b^{3} c e}{3}\right ) + x^{11} \left (\frac{6 a^{2} c^{2} f}{11} + \frac{12 a b^{2} c f}{11} + \frac{12 a b c^{2} d}{11} + \frac{b^{4} f}{11} + \frac{4 b^{3} c d}{11}\right ) + x^{10} \left (\frac{3 a^{2} c^{2} e}{5} + \frac{6 a b^{2} c e}{5} + \frac{b^{4} e}{10}\right ) + x^{9} \left (\frac{4 a^{2} b c f}{3} + \frac{2 a^{2} c^{2} d}{3} + \frac{4 a b^{3} f}{9} + \frac{4 a b^{2} c d}{3} + \frac{b^{4} d}{9}\right ) + x^{8} \left (\frac{3 a^{2} b c e}{2} + \frac{a b^{3} e}{2}\right ) + x^{7} \left (\frac{4 a^{3} c f}{7} + \frac{6 a^{2} b^{2} f}{7} + \frac{12 a^{2} b c d}{7} + \frac{4 a b^{3} d}{7}\right ) + x^{6} \left (\frac{2 a^{3} c e}{3} + a^{2} b^{2} e\right ) + x^{5} \left (\frac{4 a^{3} b f}{5} + \frac{4 a^{3} c d}{5} + \frac{6 a^{2} b^{2} d}{5}\right ) + x^{3} \left (\frac{a^{4} f}{3} + \frac{4 a^{3} b d}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**3*(a*d+a*e*x+(a*f+b*d)*x**2+b*e*x**3+(b*f+c*d)*x**4+c*e*x**5+c*f*x**6),x)

[Out]

a**4*d*x + a**4*e*x**2/2 + a**3*b*e*x**4 + b*c**3*e*x**16/4 + c**4*e*x**18/18 +
c**4*f*x**19/19 + x**17*(4*b*c**3*f/17 + c**4*d/17) + x**15*(4*a*c**3*f/15 + 2*b
**2*c**2*f/5 + 4*b*c**3*d/15) + x**14*(2*a*c**3*e/7 + 3*b**2*c**2*e/7) + x**13*(
12*a*b*c**2*f/13 + 4*a*c**3*d/13 + 4*b**3*c*f/13 + 6*b**2*c**2*d/13) + x**12*(a*
b*c**2*e + b**3*c*e/3) + x**11*(6*a**2*c**2*f/11 + 12*a*b**2*c*f/11 + 12*a*b*c**
2*d/11 + b**4*f/11 + 4*b**3*c*d/11) + x**10*(3*a**2*c**2*e/5 + 6*a*b**2*c*e/5 +
b**4*e/10) + x**9*(4*a**2*b*c*f/3 + 2*a**2*c**2*d/3 + 4*a*b**3*f/9 + 4*a*b**2*c*
d/3 + b**4*d/9) + x**8*(3*a**2*b*c*e/2 + a*b**3*e/2) + x**7*(4*a**3*c*f/7 + 6*a*
*2*b**2*f/7 + 12*a**2*b*c*d/7 + 4*a*b**3*d/7) + x**6*(2*a**3*c*e/3 + a**2*b**2*e
) + x**5*(4*a**3*b*f/5 + 4*a**3*c*d/5 + 6*a**2*b**2*d/5) + x**3*(a**4*f/3 + 4*a*
*3*b*d/3)

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GIAC/XCAS [A]  time = 0.30358, size = 645, normalized size = 1.55 \[ \frac{1}{19} \, c^{4} f x^{19} + \frac{1}{18} \, c^{4} x^{18} e + \frac{1}{17} \, c^{4} d x^{17} + \frac{4}{17} \, b c^{3} f x^{17} + \frac{1}{4} \, b c^{3} x^{16} e + \frac{4}{15} \, b c^{3} d x^{15} + \frac{2}{5} \, b^{2} c^{2} f x^{15} + \frac{4}{15} \, a c^{3} f x^{15} + \frac{3}{7} \, b^{2} c^{2} x^{14} e + \frac{2}{7} \, a c^{3} x^{14} e + \frac{6}{13} \, b^{2} c^{2} d x^{13} + \frac{4}{13} \, a c^{3} d x^{13} + \frac{4}{13} \, b^{3} c f x^{13} + \frac{12}{13} \, a b c^{2} f x^{13} + \frac{1}{3} \, b^{3} c x^{12} e + a b c^{2} x^{12} e + \frac{4}{11} \, b^{3} c d x^{11} + \frac{12}{11} \, a b c^{2} d x^{11} + \frac{1}{11} \, b^{4} f x^{11} + \frac{12}{11} \, a b^{2} c f x^{11} + \frac{6}{11} \, a^{2} c^{2} f x^{11} + \frac{1}{10} \, b^{4} x^{10} e + \frac{6}{5} \, a b^{2} c x^{10} e + \frac{3}{5} \, a^{2} c^{2} x^{10} e + \frac{1}{9} \, b^{4} d x^{9} + \frac{4}{3} \, a b^{2} c d x^{9} + \frac{2}{3} \, a^{2} c^{2} d x^{9} + \frac{4}{9} \, a b^{3} f x^{9} + \frac{4}{3} \, a^{2} b c f x^{9} + \frac{1}{2} \, a b^{3} x^{8} e + \frac{3}{2} \, a^{2} b c x^{8} e + \frac{4}{7} \, a b^{3} d x^{7} + \frac{12}{7} \, a^{2} b c d x^{7} + \frac{6}{7} \, a^{2} b^{2} f x^{7} + \frac{4}{7} \, a^{3} c f x^{7} + a^{2} b^{2} x^{6} e + \frac{2}{3} \, a^{3} c x^{6} e + \frac{6}{5} \, a^{2} b^{2} d x^{5} + \frac{4}{5} \, a^{3} c d x^{5} + \frac{4}{5} \, a^{3} b f x^{5} + a^{3} b x^{4} e + \frac{4}{3} \, a^{3} b d x^{3} + \frac{1}{3} \, a^{4} f x^{3} + \frac{1}{2} \, a^{4} x^{2} e + a^{4} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*f*x^6 + c*e*x^5 + b*e*x^3 + (c*d + b*f)*x^4 + a*e*x + (b*d + a*f)*x^2 + a*d)*(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")

[Out]

1/19*c^4*f*x^19 + 1/18*c^4*x^18*e + 1/17*c^4*d*x^17 + 4/17*b*c^3*f*x^17 + 1/4*b*
c^3*x^16*e + 4/15*b*c^3*d*x^15 + 2/5*b^2*c^2*f*x^15 + 4/15*a*c^3*f*x^15 + 3/7*b^
2*c^2*x^14*e + 2/7*a*c^3*x^14*e + 6/13*b^2*c^2*d*x^13 + 4/13*a*c^3*d*x^13 + 4/13
*b^3*c*f*x^13 + 12/13*a*b*c^2*f*x^13 + 1/3*b^3*c*x^12*e + a*b*c^2*x^12*e + 4/11*
b^3*c*d*x^11 + 12/11*a*b*c^2*d*x^11 + 1/11*b^4*f*x^11 + 12/11*a*b^2*c*f*x^11 + 6
/11*a^2*c^2*f*x^11 + 1/10*b^4*x^10*e + 6/5*a*b^2*c*x^10*e + 3/5*a^2*c^2*x^10*e +
1/9*b^4*d*x^9 + 4/3*a*b^2*c*d*x^9 + 2/3*a^2*c^2*d*x^9 + 4/9*a*b^3*f*x^9 + 4/3*a
^2*b*c*f*x^9 + 1/2*a*b^3*x^8*e + 3/2*a^2*b*c*x^8*e + 4/7*a*b^3*d*x^7 + 12/7*a^2*
b*c*d*x^7 + 6/7*a^2*b^2*f*x^7 + 4/7*a^3*c*f*x^7 + a^2*b^2*x^6*e + 2/3*a^3*c*x^6*
e + 6/5*a^2*b^2*d*x^5 + 4/5*a^3*c*d*x^5 + 4/5*a^3*b*f*x^5 + a^3*b*x^4*e + 4/3*a^
3*b*d*x^3 + 1/3*a^4*f*x^3 + 1/2*a^4*x^2*e + a^4*d*x