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

Optimal. Leaf size=159 \[ \frac{a^3 f^3 (d+e x)^4}{4 e}+\frac{a^2 b f^3 (d+e x)^6}{2 e}+\frac{c f^3 \left (a c+b^2\right ) (d+e x)^{12}}{4 e}+\frac{b f^3 \left (6 a c+b^2\right ) (d+e x)^{10}}{10 e}+\frac{3 a f^3 \left (a c+b^2\right ) (d+e x)^8}{8 e}+\frac{3 b c^2 f^3 (d+e x)^{14}}{14 e}+\frac{c^3 f^3 (d+e x)^{16}}{16 e} \]

[Out]

(a^3*f^3*(d + e*x)^4)/(4*e) + (a^2*b*f^3*(d + e*x)^6)/(2*e) + (3*a*(b^2 + a*c)*f
^3*(d + e*x)^8)/(8*e) + (b*(b^2 + 6*a*c)*f^3*(d + e*x)^10)/(10*e) + (c*(b^2 + a*
c)*f^3*(d + e*x)^12)/(4*e) + (3*b*c^2*f^3*(d + e*x)^14)/(14*e) + (c^3*f^3*(d + e
*x)^16)/(16*e)

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Rubi [A]  time = 0.757957, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a^3 f^3 (d+e x)^4}{4 e}+\frac{a^2 b f^3 (d+e x)^6}{2 e}+\frac{c f^3 \left (a c+b^2\right ) (d+e x)^{12}}{4 e}+\frac{b f^3 \left (6 a c+b^2\right ) (d+e x)^{10}}{10 e}+\frac{3 a f^3 \left (a c+b^2\right ) (d+e x)^8}{8 e}+\frac{3 b c^2 f^3 (d+e x)^{14}}{14 e}+\frac{c^3 f^3 (d+e x)^{16}}{16 e} \]

Antiderivative was successfully verified.

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

[Out]

(a^3*f^3*(d + e*x)^4)/(4*e) + (a^2*b*f^3*(d + e*x)^6)/(2*e) + (3*a*(b^2 + a*c)*f
^3*(d + e*x)^8)/(8*e) + (b*(b^2 + 6*a*c)*f^3*(d + e*x)^10)/(10*e) + (c*(b^2 + a*
c)*f^3*(d + e*x)^12)/(4*e) + (3*b*c^2*f^3*(d + e*x)^14)/(14*e) + (c^3*f^3*(d + e
*x)^16)/(16*e)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*f**3*Integral(x, (x, (d + e*x)**2))/(2*e) + a**2*b*f**3*(d + e*x)**6/(2*e)
+ 3*a*f**3*(d + e*x)**8*(a*c + b**2)/(8*e) + 3*b*c**2*f**3*(d + e*x)**14/(14*e)
+ b*f**3*(d + e*x)**10*(6*a*c + b**2)/(10*e) + c**3*f**3*(d + e*x)**16/(16*e) +
c*f**3*(d + e*x)**12*(a*c + b**2)/(4*e)

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Mathematica [B]  time = 0.639177, size = 801, normalized size = 5.04 \[ f^3 \left (\frac{1}{16} c^3 e^{15} x^{16}+c^3 d e^{14} x^{15}+\frac{3}{14} c^2 \left (35 c d^2+b\right ) e^{13} x^{14}+c^2 d \left (35 c d^2+3 b\right ) e^{12} x^{13}+\frac{1}{4} c \left (455 c^2 d^4+78 b c d^2+b^2+a c\right ) e^{11} x^{12}+3 c d \left (91 c^2 d^4+26 b c d^2+b^2+a c\right ) e^{10} x^{11}+\frac{1}{10} \left (5005 c^3 d^6+2145 b c^2 d^4+165 a c^2 d^2+165 b^2 c d^2+b^3+6 a b c\right ) e^9 x^{10}+d \left (715 c^3 d^6+429 b c^2 d^4+55 a c^2 d^2+55 b^2 c d^2+b^3+6 a b c\right ) e^8 x^9+\frac{3}{8} \left (2145 c^3 d^8+1716 b c^2 d^6+330 a c^2 d^4+330 b^2 c d^4+12 b^3 d^2+72 a b c d^2+a b^2+a^2 c\right ) e^7 x^8+\frac{1}{7} d \left (5005 c^3 d^8+5148 b c^2 d^6+1386 a c^2 d^4+1386 b^2 c d^4+84 b^3 d^2+504 a b c d^2+21 a b^2+21 a^2 c\right ) e^6 x^7+\frac{1}{2} \left (1001 c^3 d^{10}+1287 b c^2 d^8+462 a c^2 d^6+462 b^2 c d^6+42 b^3 d^4+252 a b c d^4+21 a b^2 d^2+21 a^2 c d^2+a^2 b\right ) e^5 x^6+\frac{3}{5} d \left (455 c^3 d^{10}+715 b c^2 d^8+330 a c^2 d^6+330 b^2 c d^6+42 b^3 d^4+252 a b c d^4+35 a b^2 d^2+35 a^2 c d^2+5 a^2 b\right ) e^4 x^5+\frac{1}{4} \left (455 c^3 d^{12}+858 b c^2 d^{10}+495 a c^2 d^8+495 b^2 c d^8+84 b^3 d^6+504 a b c d^6+105 a b^2 d^4+105 a^2 c d^4+30 a^2 b d^2+a^3\right ) e^3 x^4+d \left (35 c^3 d^{12}+78 b c^2 d^{10}+55 a c^2 d^8+55 b^2 c d^8+12 b^3 d^6+72 a b c d^6+21 a b^2 d^4+21 a^2 c d^4+10 a^2 b d^2+a^3\right ) e^2 x^3+\frac{3}{2} d^2 \left (c d^4+b d^2+a\right )^2 \left (5 c d^4+3 b d^2+a\right ) e x^2+d^3 \left (c d^4+b d^2+a\right )^3 x\right ) \]

Antiderivative was successfully verified.

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

[Out]

f^3*(d^3*(a + b*d^2 + c*d^4)^3*x + (3*d^2*(a + b*d^2 + c*d^4)^2*(a + 3*b*d^2 + 5
*c*d^4)*e*x^2)/2 + d*(a^3 + 10*a^2*b*d^2 + 21*a*b^2*d^4 + 21*a^2*c*d^4 + 12*b^3*
d^6 + 72*a*b*c*d^6 + 55*b^2*c*d^8 + 55*a*c^2*d^8 + 78*b*c^2*d^10 + 35*c^3*d^12)*
e^2*x^3 + ((a^3 + 30*a^2*b*d^2 + 105*a*b^2*d^4 + 105*a^2*c*d^4 + 84*b^3*d^6 + 50
4*a*b*c*d^6 + 495*b^2*c*d^8 + 495*a*c^2*d^8 + 858*b*c^2*d^10 + 455*c^3*d^12)*e^3
*x^4)/4 + (3*d*(5*a^2*b + 35*a*b^2*d^2 + 35*a^2*c*d^2 + 42*b^3*d^4 + 252*a*b*c*d
^4 + 330*b^2*c*d^6 + 330*a*c^2*d^6 + 715*b*c^2*d^8 + 455*c^3*d^10)*e^4*x^5)/5 +
((a^2*b + 21*a*b^2*d^2 + 21*a^2*c*d^2 + 42*b^3*d^4 + 252*a*b*c*d^4 + 462*b^2*c*d
^6 + 462*a*c^2*d^6 + 1287*b*c^2*d^8 + 1001*c^3*d^10)*e^5*x^6)/2 + (d*(21*a*b^2 +
21*a^2*c + 84*b^3*d^2 + 504*a*b*c*d^2 + 1386*b^2*c*d^4 + 1386*a*c^2*d^4 + 5148*
b*c^2*d^6 + 5005*c^3*d^8)*e^6*x^7)/7 + (3*(a*b^2 + a^2*c + 12*b^3*d^2 + 72*a*b*c
*d^2 + 330*b^2*c*d^4 + 330*a*c^2*d^4 + 1716*b*c^2*d^6 + 2145*c^3*d^8)*e^7*x^8)/8
+ d*(b^3 + 6*a*b*c + 55*b^2*c*d^2 + 55*a*c^2*d^2 + 429*b*c^2*d^4 + 715*c^3*d^6)
*e^8*x^9 + ((b^3 + 6*a*b*c + 165*b^2*c*d^2 + 165*a*c^2*d^2 + 2145*b*c^2*d^4 + 50
05*c^3*d^6)*e^9*x^10)/10 + 3*c*d*(b^2 + a*c + 26*b*c*d^2 + 91*c^2*d^4)*e^10*x^11
+ (c*(b^2 + a*c + 78*b*c*d^2 + 455*c^2*d^4)*e^11*x^12)/4 + c^2*d*(3*b + 35*c*d^
2)*e^12*x^13 + (3*c^2*(b + 35*c*d^2)*e^13*x^14)/14 + c^3*d*e^14*x^15 + (c^3*e^15
*x^16)/16)

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Maple [B]  time = 0.002, size = 7697, normalized size = 48.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [A]  time = 0.75888, size = 1242, normalized size = 7.81 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*c^3*e^15*f^3*x^16 + c^3*d*e^14*f^3*x^15 + 3/14*(35*c^3*d^2 + b*c^2)*e^13*f^
3*x^14 + (35*c^3*d^3 + 3*b*c^2*d)*e^12*f^3*x^13 + 1/4*(455*c^3*d^4 + 78*b*c^2*d^
2 + b^2*c + a*c^2)*e^11*f^3*x^12 + 3*(91*c^3*d^5 + 26*b*c^2*d^3 + (b^2*c + a*c^2
)*d)*e^10*f^3*x^11 + 1/10*(5005*c^3*d^6 + 2145*b*c^2*d^4 + b^3 + 6*a*b*c + 165*(
b^2*c + a*c^2)*d^2)*e^9*f^3*x^10 + (715*c^3*d^7 + 429*b*c^2*d^5 + 55*(b^2*c + a*
c^2)*d^3 + (b^3 + 6*a*b*c)*d)*e^8*f^3*x^9 + 3/8*(2145*c^3*d^8 + 1716*b*c^2*d^6 +
330*(b^2*c + a*c^2)*d^4 + a*b^2 + a^2*c + 12*(b^3 + 6*a*b*c)*d^2)*e^7*f^3*x^8 +
1/7*(5005*c^3*d^9 + 5148*b*c^2*d^7 + 1386*(b^2*c + a*c^2)*d^5 + 84*(b^3 + 6*a*b
*c)*d^3 + 21*(a*b^2 + a^2*c)*d)*e^6*f^3*x^7 + 1/2*(1001*c^3*d^10 + 1287*b*c^2*d^
8 + 462*(b^2*c + a*c^2)*d^6 + 42*(b^3 + 6*a*b*c)*d^4 + a^2*b + 21*(a*b^2 + a^2*c
)*d^2)*e^5*f^3*x^6 + 3/5*(455*c^3*d^11 + 715*b*c^2*d^9 + 330*(b^2*c + a*c^2)*d^7
+ 42*(b^3 + 6*a*b*c)*d^5 + 5*a^2*b*d + 35*(a*b^2 + a^2*c)*d^3)*e^4*f^3*x^5 + 1/
4*(455*c^3*d^12 + 858*b*c^2*d^10 + 495*(b^2*c + a*c^2)*d^8 + 84*(b^3 + 6*a*b*c)*
d^6 + 30*a^2*b*d^2 + 105*(a*b^2 + a^2*c)*d^4 + a^3)*e^3*f^3*x^4 + (35*c^3*d^13 +
78*b*c^2*d^11 + 55*(b^2*c + a*c^2)*d^9 + 12*(b^3 + 6*a*b*c)*d^7 + 10*a^2*b*d^3
+ 21*(a*b^2 + a^2*c)*d^5 + a^3*d)*e^2*f^3*x^3 + 3/2*(5*c^3*d^14 + 13*b*c^2*d^12
+ 11*(b^2*c + a*c^2)*d^10 + 3*(b^3 + 6*a*b*c)*d^8 + 5*a^2*b*d^4 + 7*(a*b^2 + a^2
*c)*d^6 + a^3*d^2)*e*f^3*x^2 + (c^3*d^15 + 3*b*c^2*d^13 + 3*(b^2*c + a*c^2)*d^11
+ (b^3 + 6*a*b*c)*d^9 + 3*a^2*b*d^5 + 3*(a*b^2 + a^2*c)*d^7 + a^3*d^3)*f^3*x

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Fricas [A]  time = 0.25855, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*x^16*f^3*e^15*c^3 + x^15*f^3*e^14*d*c^3 + 15/2*x^14*f^3*e^13*d^2*c^3 + 35*x
^13*f^3*e^12*d^3*c^3 + 455/4*x^12*f^3*e^11*d^4*c^3 + 3/14*x^14*f^3*e^13*c^2*b +
273*x^11*f^3*e^10*d^5*c^3 + 3*x^13*f^3*e^12*d*c^2*b + 1001/2*x^10*f^3*e^9*d^6*c^
3 + 39/2*x^12*f^3*e^11*d^2*c^2*b + 715*x^9*f^3*e^8*d^7*c^3 + 78*x^11*f^3*e^10*d^
3*c^2*b + 6435/8*x^8*f^3*e^7*d^8*c^3 + 429/2*x^10*f^3*e^9*d^4*c^2*b + 1/4*x^12*f
^3*e^11*c*b^2 + 1/4*x^12*f^3*e^11*c^2*a + 715*x^7*f^3*e^6*d^9*c^3 + 429*x^9*f^3*
e^8*d^5*c^2*b + 3*x^11*f^3*e^10*d*c*b^2 + 3*x^11*f^3*e^10*d*c^2*a + 1001/2*x^6*f
^3*e^5*d^10*c^3 + 1287/2*x^8*f^3*e^7*d^6*c^2*b + 33/2*x^10*f^3*e^9*d^2*c*b^2 + 3
3/2*x^10*f^3*e^9*d^2*c^2*a + 273*x^5*f^3*e^4*d^11*c^3 + 5148/7*x^7*f^3*e^6*d^7*c
^2*b + 55*x^9*f^3*e^8*d^3*c*b^2 + 55*x^9*f^3*e^8*d^3*c^2*a + 455/4*x^4*f^3*e^3*d
^12*c^3 + 1287/2*x^6*f^3*e^5*d^8*c^2*b + 495/4*x^8*f^3*e^7*d^4*c*b^2 + 1/10*x^10
*f^3*e^9*b^3 + 495/4*x^8*f^3*e^7*d^4*c^2*a + 3/5*x^10*f^3*e^9*c*b*a + 35*x^3*f^3
*e^2*d^13*c^3 + 429*x^5*f^3*e^4*d^9*c^2*b + 198*x^7*f^3*e^6*d^5*c*b^2 + x^9*f^3*
e^8*d*b^3 + 198*x^7*f^3*e^6*d^5*c^2*a + 6*x^9*f^3*e^8*d*c*b*a + 15/2*x^2*f^3*e*d
^14*c^3 + 429/2*x^4*f^3*e^3*d^10*c^2*b + 231*x^6*f^3*e^5*d^6*c*b^2 + 9/2*x^8*f^3
*e^7*d^2*b^3 + 231*x^6*f^3*e^5*d^6*c^2*a + 27*x^8*f^3*e^7*d^2*c*b*a + x*f^3*d^15
*c^3 + 78*x^3*f^3*e^2*d^11*c^2*b + 198*x^5*f^3*e^4*d^7*c*b^2 + 12*x^7*f^3*e^6*d^
3*b^3 + 198*x^5*f^3*e^4*d^7*c^2*a + 72*x^7*f^3*e^6*d^3*c*b*a + 39/2*x^2*f^3*e*d^
12*c^2*b + 495/4*x^4*f^3*e^3*d^8*c*b^2 + 21*x^6*f^3*e^5*d^4*b^3 + 495/4*x^4*f^3*
e^3*d^8*c^2*a + 126*x^6*f^3*e^5*d^4*c*b*a + 3/8*x^8*f^3*e^7*b^2*a + 3/8*x^8*f^3*
e^7*c*a^2 + 3*x*f^3*d^13*c^2*b + 55*x^3*f^3*e^2*d^9*c*b^2 + 126/5*x^5*f^3*e^4*d^
5*b^3 + 55*x^3*f^3*e^2*d^9*c^2*a + 756/5*x^5*f^3*e^4*d^5*c*b*a + 3*x^7*f^3*e^6*d
*b^2*a + 3*x^7*f^3*e^6*d*c*a^2 + 33/2*x^2*f^3*e*d^10*c*b^2 + 21*x^4*f^3*e^3*d^6*
b^3 + 33/2*x^2*f^3*e*d^10*c^2*a + 126*x^4*f^3*e^3*d^6*c*b*a + 21/2*x^6*f^3*e^5*d
^2*b^2*a + 21/2*x^6*f^3*e^5*d^2*c*a^2 + 3*x*f^3*d^11*c*b^2 + 12*x^3*f^3*e^2*d^7*
b^3 + 3*x*f^3*d^11*c^2*a + 72*x^3*f^3*e^2*d^7*c*b*a + 21*x^5*f^3*e^4*d^3*b^2*a +
21*x^5*f^3*e^4*d^3*c*a^2 + 9/2*x^2*f^3*e*d^8*b^3 + 27*x^2*f^3*e*d^8*c*b*a + 105
/4*x^4*f^3*e^3*d^4*b^2*a + 105/4*x^4*f^3*e^3*d^4*c*a^2 + 1/2*x^6*f^3*e^5*b*a^2 +
x*f^3*d^9*b^3 + 6*x*f^3*d^9*c*b*a + 21*x^3*f^3*e^2*d^5*b^2*a + 21*x^3*f^3*e^2*d
^5*c*a^2 + 3*x^5*f^3*e^4*d*b*a^2 + 21/2*x^2*f^3*e*d^6*b^2*a + 21/2*x^2*f^3*e*d^6
*c*a^2 + 15/2*x^4*f^3*e^3*d^2*b*a^2 + 3*x*f^3*d^7*b^2*a + 3*x*f^3*d^7*c*a^2 + 10
*x^3*f^3*e^2*d^3*b*a^2 + 15/2*x^2*f^3*e*d^4*b*a^2 + 1/4*x^4*f^3*e^3*a^3 + 3*x*f^
3*d^5*b*a^2 + x^3*f^3*e^2*d*a^3 + 3/2*x^2*f^3*e*d^2*a^3 + x*f^3*d^3*a^3

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Sympy [A]  time = 1.05511, size = 1654, normalized size = 10.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**3*d*e**14*f**3*x**15 + c**3*e**15*f**3*x**16/16 + x**14*(3*b*c**2*e**13*f**3/
14 + 15*c**3*d**2*e**13*f**3/2) + x**13*(3*b*c**2*d*e**12*f**3 + 35*c**3*d**3*e*
*12*f**3) + x**12*(a*c**2*e**11*f**3/4 + b**2*c*e**11*f**3/4 + 39*b*c**2*d**2*e*
*11*f**3/2 + 455*c**3*d**4*e**11*f**3/4) + x**11*(3*a*c**2*d*e**10*f**3 + 3*b**2
*c*d*e**10*f**3 + 78*b*c**2*d**3*e**10*f**3 + 273*c**3*d**5*e**10*f**3) + x**10*
(3*a*b*c*e**9*f**3/5 + 33*a*c**2*d**2*e**9*f**3/2 + b**3*e**9*f**3/10 + 33*b**2*
c*d**2*e**9*f**3/2 + 429*b*c**2*d**4*e**9*f**3/2 + 1001*c**3*d**6*e**9*f**3/2) +
x**9*(6*a*b*c*d*e**8*f**3 + 55*a*c**2*d**3*e**8*f**3 + b**3*d*e**8*f**3 + 55*b*
*2*c*d**3*e**8*f**3 + 429*b*c**2*d**5*e**8*f**3 + 715*c**3*d**7*e**8*f**3) + x**
8*(3*a**2*c*e**7*f**3/8 + 3*a*b**2*e**7*f**3/8 + 27*a*b*c*d**2*e**7*f**3 + 495*a
*c**2*d**4*e**7*f**3/4 + 9*b**3*d**2*e**7*f**3/2 + 495*b**2*c*d**4*e**7*f**3/4 +
1287*b*c**2*d**6*e**7*f**3/2 + 6435*c**3*d**8*e**7*f**3/8) + x**7*(3*a**2*c*d*e
**6*f**3 + 3*a*b**2*d*e**6*f**3 + 72*a*b*c*d**3*e**6*f**3 + 198*a*c**2*d**5*e**6
*f**3 + 12*b**3*d**3*e**6*f**3 + 198*b**2*c*d**5*e**6*f**3 + 5148*b*c**2*d**7*e*
*6*f**3/7 + 715*c**3*d**9*e**6*f**3) + x**6*(a**2*b*e**5*f**3/2 + 21*a**2*c*d**2
*e**5*f**3/2 + 21*a*b**2*d**2*e**5*f**3/2 + 126*a*b*c*d**4*e**5*f**3 + 231*a*c**
2*d**6*e**5*f**3 + 21*b**3*d**4*e**5*f**3 + 231*b**2*c*d**6*e**5*f**3 + 1287*b*c
**2*d**8*e**5*f**3/2 + 1001*c**3*d**10*e**5*f**3/2) + x**5*(3*a**2*b*d*e**4*f**3
+ 21*a**2*c*d**3*e**4*f**3 + 21*a*b**2*d**3*e**4*f**3 + 756*a*b*c*d**5*e**4*f**
3/5 + 198*a*c**2*d**7*e**4*f**3 + 126*b**3*d**5*e**4*f**3/5 + 198*b**2*c*d**7*e*
*4*f**3 + 429*b*c**2*d**9*e**4*f**3 + 273*c**3*d**11*e**4*f**3) + x**4*(a**3*e**
3*f**3/4 + 15*a**2*b*d**2*e**3*f**3/2 + 105*a**2*c*d**4*e**3*f**3/4 + 105*a*b**2
*d**4*e**3*f**3/4 + 126*a*b*c*d**6*e**3*f**3 + 495*a*c**2*d**8*e**3*f**3/4 + 21*
b**3*d**6*e**3*f**3 + 495*b**2*c*d**8*e**3*f**3/4 + 429*b*c**2*d**10*e**3*f**3/2
+ 455*c**3*d**12*e**3*f**3/4) + x**3*(a**3*d*e**2*f**3 + 10*a**2*b*d**3*e**2*f*
*3 + 21*a**2*c*d**5*e**2*f**3 + 21*a*b**2*d**5*e**2*f**3 + 72*a*b*c*d**7*e**2*f*
*3 + 55*a*c**2*d**9*e**2*f**3 + 12*b**3*d**7*e**2*f**3 + 55*b**2*c*d**9*e**2*f**
3 + 78*b*c**2*d**11*e**2*f**3 + 35*c**3*d**13*e**2*f**3) + x**2*(3*a**3*d**2*e*f
**3/2 + 15*a**2*b*d**4*e*f**3/2 + 21*a**2*c*d**6*e*f**3/2 + 21*a*b**2*d**6*e*f**
3/2 + 27*a*b*c*d**8*e*f**3 + 33*a*c**2*d**10*e*f**3/2 + 9*b**3*d**8*e*f**3/2 + 3
3*b**2*c*d**10*e*f**3/2 + 39*b*c**2*d**12*e*f**3/2 + 15*c**3*d**14*e*f**3/2) + x
*(a**3*d**3*f**3 + 3*a**2*b*d**5*f**3 + 3*a**2*c*d**7*f**3 + 3*a*b**2*d**7*f**3
+ 6*a*b*c*d**9*f**3 + 3*a*c**2*d**11*f**3 + b**3*d**9*f**3 + 3*b**2*c*d**11*f**3
+ 3*b*c**2*d**13*f**3 + c**3*d**15*f**3)

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GIAC/XCAS [A]  time = 0.27078, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done