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

Optimal. Leaf size=361 \[ \frac{3 d f (a+b x)^8 \left (5 a^2 d^2 f^2-5 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )}{8 b^7}+\frac{(a+b x)^7 (-2 a d f+b c f+b d e) \left (10 a^2 d^2 f^2-10 a b d f (c f+d e)+b^2 \left (c^2 f^2+8 c d e f+d^2 e^2\right )\right )}{7 b^7}+\frac{(a+b x)^6 (b c-a d) (b e-a f) \left (5 a^2 d^2 f^2-5 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )}{2 b^7}+\frac{d^2 f^2 (a+b x)^9 (-2 a d f+b c f+b d e)}{3 b^7}+\frac{3 (a+b x)^5 (b c-a d)^2 (b e-a f)^2 (-2 a d f+b c f+b d e)}{5 b^7}+\frac{(a+b x)^4 (b c-a d)^3 (b e-a f)^3}{4 b^7}+\frac{d^3 f^3 (a+b x)^{10}}{10 b^7} \]

[Out]

((b*c - a*d)^3*(b*e - a*f)^3*(a + b*x)^4)/(4*b^7) + (3*(b*c - a*d)^2*(b*e - a*f)
^2*(b*d*e + b*c*f - 2*a*d*f)*(a + b*x)^5)/(5*b^7) + ((b*c - a*d)*(b*e - a*f)*(5*
a^2*d^2*f^2 - 5*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*(a +
b*x)^6)/(2*b^7) + ((b*d*e + b*c*f - 2*a*d*f)*(10*a^2*d^2*f^2 - 10*a*b*d*f*(d*e +
c*f) + b^2*(d^2*e^2 + 8*c*d*e*f + c^2*f^2))*(a + b*x)^7)/(7*b^7) + (3*d*f*(5*a^
2*d^2*f^2 - 5*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*(a + b*
x)^8)/(8*b^7) + (d^2*f^2*(b*d*e + b*c*f - 2*a*d*f)*(a + b*x)^9)/(3*b^7) + (d^3*f
^3*(a + b*x)^10)/(10*b^7)

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Rubi [A]  time = 1.79802, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{3 d f (a+b x)^8 \left (5 a^2 d^2 f^2-5 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )}{8 b^7}+\frac{(a+b x)^7 (-2 a d f+b c f+b d e) \left (10 a^2 d^2 f^2-10 a b d f (c f+d e)+b^2 \left (c^2 f^2+8 c d e f+d^2 e^2\right )\right )}{7 b^7}+\frac{(a+b x)^6 (b c-a d) (b e-a f) \left (5 a^2 d^2 f^2-5 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )}{2 b^7}+\frac{d^2 f^2 (a+b x)^9 (-2 a d f+b c f+b d e)}{3 b^7}+\frac{3 (a+b x)^5 (b c-a d)^2 (b e-a f)^2 (-2 a d f+b c f+b d e)}{5 b^7}+\frac{(a+b x)^4 (b c-a d)^3 (b e-a f)^3}{4 b^7}+\frac{d^3 f^3 (a+b x)^{10}}{10 b^7} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^3*(b*e - a*f)^3*(a + b*x)^4)/(4*b^7) + (3*(b*c - a*d)^2*(b*e - a*f)
^2*(b*d*e + b*c*f - 2*a*d*f)*(a + b*x)^5)/(5*b^7) + ((b*c - a*d)*(b*e - a*f)*(5*
a^2*d^2*f^2 - 5*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*(a +
b*x)^6)/(2*b^7) + ((b*d*e + b*c*f - 2*a*d*f)*(10*a^2*d^2*f^2 - 10*a*b*d*f*(d*e +
c*f) + b^2*(d^2*e^2 + 8*c*d*e*f + c^2*f^2))*(a + b*x)^7)/(7*b^7) + (3*d*f*(5*a^
2*d^2*f^2 - 5*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*(a + b*
x)^8)/(8*b^7) + (d^2*f^2*(b*d*e + b*c*f - 2*a*d*f)*(a + b*x)^9)/(3*b^7) + (d^3*f
^3*(a + b*x)^10)/(10*b^7)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.470189, size = 653, normalized size = 1.81 \[ a^3 c^3 e^3 x+\frac{3}{8} b d f x^8 \left (a^2 d^2 f^2+3 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+a c e x^3 \left (a^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )+3 a b c e (c f+d e)+b^2 c^2 e^2\right )+\frac{3}{2} a^2 c^2 e^2 x^2 (a c f+a d e+b c e)+\frac{1}{7} x^7 \left (a^3 d^3 f^3+9 a^2 b d^2 f^2 (c f+d e)+9 a b^2 d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b^3 \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )\right )+\frac{1}{2} x^6 \left (a^3 d^2 f^2 (c f+d e)+3 a^2 b d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+a b^2 \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )+b^3 c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac{3}{5} x^5 \left (a^3 d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+a^2 b \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )+3 a b^2 c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b^3 c^2 e^2 (c f+d e)\right )+\frac{1}{4} x^4 \left (a^3 \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )+9 a^2 b c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )+9 a b^2 c^2 e^2 (c f+d e)+b^3 c^3 e^3\right )+\frac{1}{3} b^2 d^2 f^2 x^9 (a d f+b c f+b d e)+\frac{1}{10} b^3 d^3 f^3 x^{10} \]

Antiderivative was successfully verified.

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

[Out]

a^3*c^3*e^3*x + (3*a^2*c^2*e^2*(b*c*e + a*d*e + a*c*f)*x^2)/2 + a*c*e*(b^2*c^2*e
^2 + 3*a*b*c*e*(d*e + c*f) + a^2*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x^3 + ((b^3*c^
3*e^3 + 9*a*b^2*c^2*e^2*(d*e + c*f) + 9*a^2*b*c*e*(d^2*e^2 + 3*c*d*e*f + c^2*f^2
) + a^3*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3))*x^4)/4 + (3*(b^3*c^
2*e^2*(d*e + c*f) + 3*a*b^2*c*e*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + a^3*d*f*(d^2*e
^2 + 3*c*d*e*f + c^2*f^2) + a^2*b*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3
*f^3))*x^5)/5 + ((a^3*d^2*f^2*(d*e + c*f) + b^3*c*e*(d^2*e^2 + 3*c*d*e*f + c^2*f
^2) + 3*a^2*b*d*f*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + a*b^2*(d^3*e^3 + 9*c*d^2*e^2
*f + 9*c^2*d*e*f^2 + c^3*f^3))*x^6)/2 + ((a^3*d^3*f^3 + 9*a^2*b*d^2*f^2*(d*e + c
*f) + 9*a*b^2*d*f*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + b^3*(d^3*e^3 + 9*c*d^2*e^2*f
+ 9*c^2*d*e*f^2 + c^3*f^3))*x^7)/7 + (3*b*d*f*(a^2*d^2*f^2 + 3*a*b*d*f*(d*e + c
*f) + b^2*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x^8)/8 + (b^2*d^2*f^2*(b*d*e + b*c*f
+ a*d*f)*x^9)/3 + (b^3*d^3*f^3*x^10)/10

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Maple [B]  time = 0.002, size = 861, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*b^3*d^3*f^3*x^10+1/3*(a*d*f+b*c*f+b*d*e)*b^2*d^2*f^2*x^9+1/8*((a*c*f+a*d*e+
b*c*e)*b^2*d^2*f^2+2*(a*d*f+b*c*f+b*d*e)^2*b*d*f+b*d*f*(2*(a*c*f+a*d*e+b*c*e)*b*
d*f+(a*d*f+b*c*f+b*d*e)^2))*x^8+1/7*(a*c*e*b^2*d^2*f^2+2*(a*c*f+a*d*e+b*c*e)*(a*
d*f+b*c*f+b*d*e)*b*d*f+(a*d*f+b*c*f+b*d*e)*(2*(a*c*f+a*d*e+b*c*e)*b*d*f+(a*d*f+b
*c*f+b*d*e)^2)+b*d*f*(2*a*c*e*b*d*f+2*(a*c*f+a*d*e+b*c*e)*(a*d*f+b*c*f+b*d*e)))*
x^7+1/6*(2*a*c*e*(a*d*f+b*c*f+b*d*e)*b*d*f+(a*c*f+a*d*e+b*c*e)*(2*(a*c*f+a*d*e+b
*c*e)*b*d*f+(a*d*f+b*c*f+b*d*e)^2)+(a*d*f+b*c*f+b*d*e)*(2*a*c*e*b*d*f+2*(a*c*f+a
*d*e+b*c*e)*(a*d*f+b*c*f+b*d*e))+b*d*f*(2*a*c*e*(a*d*f+b*c*f+b*d*e)+(a*c*f+a*d*e
+b*c*e)^2))*x^6+1/5*(a*c*e*(2*(a*c*f+a*d*e+b*c*e)*b*d*f+(a*d*f+b*c*f+b*d*e)^2)+(
a*c*f+a*d*e+b*c*e)*(2*a*c*e*b*d*f+2*(a*c*f+a*d*e+b*c*e)*(a*d*f+b*c*f+b*d*e))+(a*
d*f+b*c*f+b*d*e)*(2*a*c*e*(a*d*f+b*c*f+b*d*e)+(a*c*f+a*d*e+b*c*e)^2)+2*b*d*f*a*c
*e*(a*c*f+a*d*e+b*c*e))*x^5+1/4*(a*c*e*(2*a*c*e*b*d*f+2*(a*c*f+a*d*e+b*c*e)*(a*d
*f+b*c*f+b*d*e))+(a*c*f+a*d*e+b*c*e)*(2*a*c*e*(a*d*f+b*c*f+b*d*e)+(a*c*f+a*d*e+b
*c*e)^2)+2*(a*d*f+b*c*f+b*d*e)*a*c*e*(a*c*f+a*d*e+b*c*e)+b*d*f*a^2*c^2*e^2)*x^4+
1/3*(a*c*e*(2*a*c*e*(a*d*f+b*c*f+b*d*e)+(a*c*f+a*d*e+b*c*e)^2)+2*(a*c*f+a*d*e+b*
c*e)^2*a*c*e+(a*d*f+b*c*f+b*d*e)*a^2*c^2*e^2)*x^3+3/2*a^2*c^2*e^2*(a*c*f+a*d*e+b
*c*e)*x^2+a^3*c^3*e^3*x

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Maxima [A]  time = 0.778591, size = 622, normalized size = 1.72 \[ \frac{1}{10} \, b^{3} d^{3} f^{3} x^{10} + \frac{1}{3} \,{\left (b d e + b c f + a d f\right )} b^{2} d^{2} f^{2} x^{9} + \frac{3}{8} \,{\left (b d e + b c f + a d f\right )}^{2} b d f x^{8} + a^{3} c^{3} e^{3} x + \frac{1}{7} \,{\left (b d e + b c f + a d f\right )}^{3} x^{7} + \frac{1}{4} \,{\left (3 \, b d f x^{4} + 4 \,{\left (b d e + b c f + a d f\right )} x^{3} + 6 \,{\left (b c e + a d e + a c f\right )} x^{2}\right )} a^{2} c^{2} e^{2} + \frac{1}{4} \,{\left (b c e + a d e + a c f\right )}^{3} x^{4} + \frac{1}{70} \,{\left (30 \, b^{2} d^{2} f^{2} x^{7} + 70 \,{\left (b d e + b c f + a d f\right )} b d f x^{6} + 42 \,{\left (b d e + b c f + a d f\right )}^{2} x^{5} + 70 \,{\left (b c e + a d e + a c f\right )}^{2} x^{3} + 21 \,{\left (4 \, b d f x^{5} + 5 \,{\left (b d e +{\left (b c + a d\right )} f\right )} x^{4}\right )}{\left (b c e + a d e + a c f\right )}\right )} a c e + \frac{1}{10} \,{\left (5 \, b d f x^{6} + 6 \,{\left (b d e +{\left (b c + a d\right )} f\right )} x^{5}\right )}{\left (b c e + a d e + a c f\right )}^{2} + \frac{1}{56} \,{\left (21 \, b^{2} d^{2} f^{2} x^{8} + 48 \,{\left (b^{2} d^{2} e f +{\left (b^{2} c d + a b d^{2}\right )} f^{2}\right )} x^{7} + 28 \,{\left (b^{2} d^{2} e^{2} + 2 \,{\left (b^{2} c d + a b d^{2}\right )} e f +{\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} f^{2}\right )} x^{6}\right )}{\left (b c e + a d e + a c f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*b^3*d^3*f^3*x^10 + 1/3*(b*d*e + b*c*f + a*d*f)*b^2*d^2*f^2*x^9 + 3/8*(b*d*e
+ b*c*f + a*d*f)^2*b*d*f*x^8 + a^3*c^3*e^3*x + 1/7*(b*d*e + b*c*f + a*d*f)^3*x^
7 + 1/4*(3*b*d*f*x^4 + 4*(b*d*e + b*c*f + a*d*f)*x^3 + 6*(b*c*e + a*d*e + a*c*f)
*x^2)*a^2*c^2*e^2 + 1/4*(b*c*e + a*d*e + a*c*f)^3*x^4 + 1/70*(30*b^2*d^2*f^2*x^7
+ 70*(b*d*e + b*c*f + a*d*f)*b*d*f*x^6 + 42*(b*d*e + b*c*f + a*d*f)^2*x^5 + 70*
(b*c*e + a*d*e + a*c*f)^2*x^3 + 21*(4*b*d*f*x^5 + 5*(b*d*e + (b*c + a*d)*f)*x^4)
*(b*c*e + a*d*e + a*c*f))*a*c*e + 1/10*(5*b*d*f*x^6 + 6*(b*d*e + (b*c + a*d)*f)*
x^5)*(b*c*e + a*d*e + a*c*f)^2 + 1/56*(21*b^2*d^2*f^2*x^8 + 48*(b^2*d^2*e*f + (b
^2*c*d + a*b*d^2)*f^2)*x^7 + 28*(b^2*d^2*e^2 + 2*(b^2*c*d + a*b*d^2)*e*f + (b^2*
c^2 + 2*a*b*c*d + a^2*d^2)*f^2)*x^6)*(b*c*e + a*d*e + a*c*f)

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Fricas [A]  time = 0.256996, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*x^10*f^3*d^3*b^3 + 1/3*x^9*f^2*e*d^3*b^3 + 1/3*x^9*f^3*d^2*c*b^3 + 1/3*x^9*
f^3*d^3*b^2*a + 3/8*x^8*f*e^2*d^3*b^3 + 9/8*x^8*f^2*e*d^2*c*b^3 + 3/8*x^8*f^3*d*
c^2*b^3 + 9/8*x^8*f^2*e*d^3*b^2*a + 9/8*x^8*f^3*d^2*c*b^2*a + 3/8*x^8*f^3*d^3*b*
a^2 + 1/7*x^7*e^3*d^3*b^3 + 9/7*x^7*f*e^2*d^2*c*b^3 + 9/7*x^7*f^2*e*d*c^2*b^3 +
1/7*x^7*f^3*c^3*b^3 + 9/7*x^7*f*e^2*d^3*b^2*a + 27/7*x^7*f^2*e*d^2*c*b^2*a + 9/7
*x^7*f^3*d*c^2*b^2*a + 9/7*x^7*f^2*e*d^3*b*a^2 + 9/7*x^7*f^3*d^2*c*b*a^2 + 1/7*x
^7*f^3*d^3*a^3 + 1/2*x^6*e^3*d^2*c*b^3 + 3/2*x^6*f*e^2*d*c^2*b^3 + 1/2*x^6*f^2*e
*c^3*b^3 + 1/2*x^6*e^3*d^3*b^2*a + 9/2*x^6*f*e^2*d^2*c*b^2*a + 9/2*x^6*f^2*e*d*c
^2*b^2*a + 1/2*x^6*f^3*c^3*b^2*a + 3/2*x^6*f*e^2*d^3*b*a^2 + 9/2*x^6*f^2*e*d^2*c
*b*a^2 + 3/2*x^6*f^3*d*c^2*b*a^2 + 1/2*x^6*f^2*e*d^3*a^3 + 1/2*x^6*f^3*d^2*c*a^3
+ 3/5*x^5*e^3*d*c^2*b^3 + 3/5*x^5*f*e^2*c^3*b^3 + 9/5*x^5*e^3*d^2*c*b^2*a + 27/
5*x^5*f*e^2*d*c^2*b^2*a + 9/5*x^5*f^2*e*c^3*b^2*a + 3/5*x^5*e^3*d^3*b*a^2 + 27/5
*x^5*f*e^2*d^2*c*b*a^2 + 27/5*x^5*f^2*e*d*c^2*b*a^2 + 3/5*x^5*f^3*c^3*b*a^2 + 3/
5*x^5*f*e^2*d^3*a^3 + 9/5*x^5*f^2*e*d^2*c*a^3 + 3/5*x^5*f^3*d*c^2*a^3 + 1/4*x^4*
e^3*c^3*b^3 + 9/4*x^4*e^3*d*c^2*b^2*a + 9/4*x^4*f*e^2*c^3*b^2*a + 9/4*x^4*e^3*d^
2*c*b*a^2 + 27/4*x^4*f*e^2*d*c^2*b*a^2 + 9/4*x^4*f^2*e*c^3*b*a^2 + 1/4*x^4*e^3*d
^3*a^3 + 9/4*x^4*f*e^2*d^2*c*a^3 + 9/4*x^4*f^2*e*d*c^2*a^3 + 1/4*x^4*f^3*c^3*a^3
+ x^3*e^3*c^3*b^2*a + 3*x^3*e^3*d*c^2*b*a^2 + 3*x^3*f*e^2*c^3*b*a^2 + x^3*e^3*d
^2*c*a^3 + 3*x^3*f*e^2*d*c^2*a^3 + x^3*f^2*e*c^3*a^3 + 3/2*x^2*e^3*c^3*b*a^2 + 3
/2*x^2*e^3*d*c^2*a^3 + 3/2*x^2*f*e^2*c^3*a^3 + x*e^3*c^3*a^3

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Sympy [A]  time = 0.65747, size = 1018, normalized size = 2.82 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*c**3*e**3*x + b**3*d**3*f**3*x**10/10 + x**9*(a*b**2*d**3*f**3/3 + b**3*c*d
**2*f**3/3 + b**3*d**3*e*f**2/3) + x**8*(3*a**2*b*d**3*f**3/8 + 9*a*b**2*c*d**2*
f**3/8 + 9*a*b**2*d**3*e*f**2/8 + 3*b**3*c**2*d*f**3/8 + 9*b**3*c*d**2*e*f**2/8
+ 3*b**3*d**3*e**2*f/8) + x**7*(a**3*d**3*f**3/7 + 9*a**2*b*c*d**2*f**3/7 + 9*a*
*2*b*d**3*e*f**2/7 + 9*a*b**2*c**2*d*f**3/7 + 27*a*b**2*c*d**2*e*f**2/7 + 9*a*b*
*2*d**3*e**2*f/7 + b**3*c**3*f**3/7 + 9*b**3*c**2*d*e*f**2/7 + 9*b**3*c*d**2*e**
2*f/7 + b**3*d**3*e**3/7) + x**6*(a**3*c*d**2*f**3/2 + a**3*d**3*e*f**2/2 + 3*a*
*2*b*c**2*d*f**3/2 + 9*a**2*b*c*d**2*e*f**2/2 + 3*a**2*b*d**3*e**2*f/2 + a*b**2*
c**3*f**3/2 + 9*a*b**2*c**2*d*e*f**2/2 + 9*a*b**2*c*d**2*e**2*f/2 + a*b**2*d**3*
e**3/2 + b**3*c**3*e*f**2/2 + 3*b**3*c**2*d*e**2*f/2 + b**3*c*d**2*e**3/2) + x**
5*(3*a**3*c**2*d*f**3/5 + 9*a**3*c*d**2*e*f**2/5 + 3*a**3*d**3*e**2*f/5 + 3*a**2
*b*c**3*f**3/5 + 27*a**2*b*c**2*d*e*f**2/5 + 27*a**2*b*c*d**2*e**2*f/5 + 3*a**2*
b*d**3*e**3/5 + 9*a*b**2*c**3*e*f**2/5 + 27*a*b**2*c**2*d*e**2*f/5 + 9*a*b**2*c*
d**2*e**3/5 + 3*b**3*c**3*e**2*f/5 + 3*b**3*c**2*d*e**3/5) + x**4*(a**3*c**3*f**
3/4 + 9*a**3*c**2*d*e*f**2/4 + 9*a**3*c*d**2*e**2*f/4 + a**3*d**3*e**3/4 + 9*a**
2*b*c**3*e*f**2/4 + 27*a**2*b*c**2*d*e**2*f/4 + 9*a**2*b*c*d**2*e**3/4 + 9*a*b**
2*c**3*e**2*f/4 + 9*a*b**2*c**2*d*e**3/4 + b**3*c**3*e**3/4) + x**3*(a**3*c**3*e
*f**2 + 3*a**3*c**2*d*e**2*f + a**3*c*d**2*e**3 + 3*a**2*b*c**3*e**2*f + 3*a**2*
b*c**2*d*e**3 + a*b**2*c**3*e**3) + x**2*(3*a**3*c**3*e**2*f/2 + 3*a**3*c**2*d*e
**3/2 + 3*a**2*b*c**3*e**3/2)

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GIAC/XCAS [A]  time = 0.258286, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done