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

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]

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

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Rubi [A]  time = 0.66, antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2059, 88} \[ \frac {3 d f (a+b x)^8 \left (5 a^2 d^2 f^2-5 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )}{8 b^7}+\frac {(a+b x)^7 (-2 a d f+b c f+b d e) \left (10 a^2 d^2 f^2-10 a b d f (c f+d e)+b^2 \left (c^2 f^2+8 c d e f+d^2 e^2\right )\right )}{7 b^7}+\frac {(a+b x)^6 (b c-a d) (b e-a f) \left (5 a^2 d^2 f^2-5 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )}{2 b^7}+\frac {d^2 f^2 (a+b x)^9 (-2 a d f+b c f+b d e)}{3 b^7}+\frac {3 (a+b x)^5 (b c-a d)^2 (b e-a f)^2 (-2 a d f+b c f+b d e)}{5 b^7}+\frac {(a+b x)^4 (b c-a d)^3 (b e-a f)^3}{4 b^7}+\frac {d^3 f^3 (a+b x)^{10}}{10 b^7} \]

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)

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2059

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[u^p, x] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x]
&& IntegerQ[p]

Rubi steps

\begin {align*} \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 &=\int (a+b x)^3 (c+d x)^3 (e+f x)^3 \, dx\\ &=\int \left (\frac {(b c-a d)^3 (b e-a f)^3 (a+b x)^3}{b^6}+\frac {3 (b c-a d)^2 (b e-a f)^2 (b d e+b c f-2 a d f) (a+b x)^4}{b^6}+\frac {3 (b c-a d) (b e-a f) \left (5 a^2 d^2 f^2-5 a b d f (d e+c f)+b^2 \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) (a+b x)^5}{b^6}+\frac {(b d e+b c f-2 a d f) \left (b^2 d^2 e^2+8 b^2 c d e f-10 a b d^2 e f+b^2 c^2 f^2-10 a b c d f^2+10 a^2 d^2 f^2\right ) (a+b x)^6}{b^6}+\frac {3 d f \left (5 a^2 d^2 f^2-5 a b d f (d e+c f)+b^2 \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) (a+b x)^7}{b^6}+\frac {3 d^2 f^2 (b d e+b c f-2 a d f) (a+b x)^8}{b^6}+\frac {d^3 f^3 (a+b x)^9}{b^6}\right ) \, dx\\ &=\frac {(b c-a d)^3 (b e-a f)^3 (a+b x)^4}{4 b^7}+\frac {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}+\frac {(b c-a d) (b e-a f) \left (5 a^2 d^2 f^2-5 a b d f (d e+c f)+b^2 \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) (a+b x)^6}{2 b^7}+\frac {(b d e+b c f-2 a d f) \left (10 a^2 d^2 f^2-10 a b d f (d e+c f)+b^2 \left (d^2 e^2+8 c d e f+c^2 f^2\right )\right ) (a+b x)^7}{7 b^7}+\frac {3 d f \left (5 a^2 d^2 f^2-5 a b d f (d e+c f)+b^2 \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) (a+b x)^8}{8 b^7}+\frac {d^2 f^2 (b d e+b c f-2 a d f) (a+b x)^9}{3 b^7}+\frac {d^3 f^3 (a+b x)^{10}}{10 b^7}\\ \end {align*}

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Mathematica [A]  time = 0.22, 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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fricas [B]  time = 0.67, size = 987, normalized size = 2.73 \[ \frac {1}{10} x^{10} f^{3} d^{3} b^{3} + \frac {1}{3} x^{9} f^{2} e d^{3} b^{3} + \frac {1}{3} x^{9} f^{3} d^{2} c b^{3} + \frac {1}{3} x^{9} f^{3} d^{3} b^{2} a + \frac {3}{8} x^{8} f e^{2} d^{3} b^{3} + \frac {9}{8} x^{8} f^{2} e d^{2} c b^{3} + \frac {3}{8} x^{8} f^{3} d c^{2} b^{3} + \frac {9}{8} x^{8} f^{2} e d^{3} b^{2} a + \frac {9}{8} x^{8} f^{3} d^{2} c b^{2} a + \frac {3}{8} x^{8} f^{3} d^{3} b a^{2} + \frac {1}{7} x^{7} e^{3} d^{3} b^{3} + \frac {9}{7} x^{7} f e^{2} d^{2} c b^{3} + \frac {9}{7} x^{7} f^{2} e d c^{2} b^{3} + \frac {1}{7} x^{7} f^{3} c^{3} b^{3} + \frac {9}{7} x^{7} f e^{2} d^{3} b^{2} a + \frac {27}{7} x^{7} f^{2} e d^{2} c b^{2} a + \frac {9}{7} x^{7} f^{3} d c^{2} b^{2} a + \frac {9}{7} x^{7} f^{2} e d^{3} b a^{2} + \frac {9}{7} x^{7} f^{3} d^{2} c b a^{2} + \frac {1}{7} x^{7} f^{3} d^{3} a^{3} + \frac {1}{2} x^{6} e^{3} d^{2} c b^{3} + \frac {3}{2} x^{6} f e^{2} d c^{2} b^{3} + \frac {1}{2} x^{6} f^{2} e c^{3} b^{3} + \frac {1}{2} x^{6} e^{3} d^{3} b^{2} a + \frac {9}{2} x^{6} f e^{2} d^{2} c b^{2} a + \frac {9}{2} x^{6} f^{2} e d c^{2} b^{2} a + \frac {1}{2} x^{6} f^{3} c^{3} b^{2} a + \frac {3}{2} x^{6} f e^{2} d^{3} b a^{2} + \frac {9}{2} x^{6} f^{2} e d^{2} c b a^{2} + \frac {3}{2} x^{6} f^{3} d c^{2} b a^{2} + \frac {1}{2} x^{6} f^{2} e d^{3} a^{3} + \frac {1}{2} x^{6} f^{3} d^{2} c a^{3} + \frac {3}{5} x^{5} e^{3} d c^{2} b^{3} + \frac {3}{5} x^{5} f e^{2} c^{3} b^{3} + \frac {9}{5} x^{5} e^{3} d^{2} c b^{2} a + \frac {27}{5} x^{5} f e^{2} d c^{2} b^{2} a + \frac {9}{5} x^{5} f^{2} e c^{3} b^{2} a + \frac {3}{5} x^{5} e^{3} d^{3} b a^{2} + \frac {27}{5} x^{5} f e^{2} d^{2} c b a^{2} + \frac {27}{5} x^{5} f^{2} e d c^{2} b a^{2} + \frac {3}{5} x^{5} f^{3} c^{3} b a^{2} + \frac {3}{5} x^{5} f e^{2} d^{3} a^{3} + \frac {9}{5} x^{5} f^{2} e d^{2} c a^{3} + \frac {3}{5} x^{5} f^{3} d c^{2} a^{3} + \frac {1}{4} x^{4} e^{3} c^{3} b^{3} + \frac {9}{4} x^{4} e^{3} d c^{2} b^{2} a + \frac {9}{4} x^{4} f e^{2} c^{3} b^{2} a + \frac {9}{4} x^{4} e^{3} d^{2} c b a^{2} + \frac {27}{4} x^{4} f e^{2} d c^{2} b a^{2} + \frac {9}{4} x^{4} f^{2} e c^{3} b a^{2} + \frac {1}{4} x^{4} e^{3} d^{3} a^{3} + \frac {9}{4} x^{4} f e^{2} d^{2} c a^{3} + \frac {9}{4} x^{4} f^{2} e d c^{2} a^{3} + \frac {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} + \frac {3}{2} x^{2} e^{3} c^{3} b a^{2} + \frac {3}{2} x^{2} e^{3} d c^{2} a^{3} + \frac {3}{2} x^{2} f e^{2} c^{3} a^{3} + x e^{3} c^{3} a^{3} \]

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, 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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giac [B]  time = 0.27, size = 971, normalized size = 2.69 \[ \frac {1}{10} \, b^{3} d^{3} f^{3} x^{10} + \frac {1}{3} \, b^{3} c d^{2} f^{3} x^{9} + \frac {1}{3} \, a b^{2} d^{3} f^{3} x^{9} + \frac {1}{3} \, b^{3} d^{3} f^{2} x^{9} e + \frac {3}{8} \, b^{3} c^{2} d f^{3} x^{8} + \frac {9}{8} \, a b^{2} c d^{2} f^{3} x^{8} + \frac {3}{8} \, a^{2} b d^{3} f^{3} x^{8} + \frac {9}{8} \, b^{3} c d^{2} f^{2} x^{8} e + \frac {9}{8} \, a b^{2} d^{3} f^{2} x^{8} e + \frac {1}{7} \, b^{3} c^{3} f^{3} x^{7} + \frac {9}{7} \, a b^{2} c^{2} d f^{3} x^{7} + \frac {9}{7} \, a^{2} b c d^{2} f^{3} x^{7} + \frac {1}{7} \, a^{3} d^{3} f^{3} x^{7} + \frac {3}{8} \, b^{3} d^{3} f x^{8} e^{2} + \frac {9}{7} \, b^{3} c^{2} d f^{2} x^{7} e + \frac {27}{7} \, a b^{2} c d^{2} f^{2} x^{7} e + \frac {9}{7} \, a^{2} b d^{3} f^{2} x^{7} e + \frac {1}{2} \, a b^{2} c^{3} f^{3} x^{6} + \frac {3}{2} \, a^{2} b c^{2} d f^{3} x^{6} + \frac {1}{2} \, a^{3} c d^{2} f^{3} x^{6} + \frac {9}{7} \, b^{3} c d^{2} f x^{7} e^{2} + \frac {9}{7} \, a b^{2} d^{3} f x^{7} e^{2} + \frac {1}{2} \, b^{3} c^{3} f^{2} x^{6} e + \frac {9}{2} \, a b^{2} c^{2} d f^{2} x^{6} e + \frac {9}{2} \, a^{2} b c d^{2} f^{2} x^{6} e + \frac {1}{2} \, a^{3} d^{3} f^{2} x^{6} e + \frac {3}{5} \, a^{2} b c^{3} f^{3} x^{5} + \frac {3}{5} \, a^{3} c^{2} d f^{3} x^{5} + \frac {1}{7} \, b^{3} d^{3} x^{7} e^{3} + \frac {3}{2} \, b^{3} c^{2} d f x^{6} e^{2} + \frac {9}{2} \, a b^{2} c d^{2} f x^{6} e^{2} + \frac {3}{2} \, a^{2} b d^{3} f x^{6} e^{2} + \frac {9}{5} \, a b^{2} c^{3} f^{2} x^{5} e + \frac {27}{5} \, a^{2} b c^{2} d f^{2} x^{5} e + \frac {9}{5} \, a^{3} c d^{2} f^{2} x^{5} e + \frac {1}{4} \, a^{3} c^{3} f^{3} x^{4} + \frac {1}{2} \, b^{3} c d^{2} x^{6} e^{3} + \frac {1}{2} \, a b^{2} d^{3} x^{6} e^{3} + \frac {3}{5} \, b^{3} c^{3} f x^{5} e^{2} + \frac {27}{5} \, a b^{2} c^{2} d f x^{5} e^{2} + \frac {27}{5} \, a^{2} b c d^{2} f x^{5} e^{2} + \frac {3}{5} \, a^{3} d^{3} f x^{5} e^{2} + \frac {9}{4} \, a^{2} b c^{3} f^{2} x^{4} e + \frac {9}{4} \, a^{3} c^{2} d f^{2} x^{4} e + \frac {3}{5} \, b^{3} c^{2} d x^{5} e^{3} + \frac {9}{5} \, a b^{2} c d^{2} x^{5} e^{3} + \frac {3}{5} \, a^{2} b d^{3} x^{5} e^{3} + \frac {9}{4} \, a b^{2} c^{3} f x^{4} e^{2} + \frac {27}{4} \, a^{2} b c^{2} d f x^{4} e^{2} + \frac {9}{4} \, a^{3} c d^{2} f x^{4} e^{2} + a^{3} c^{3} f^{2} x^{3} e + \frac {1}{4} \, b^{3} c^{3} x^{4} e^{3} + \frac {9}{4} \, a b^{2} c^{2} d x^{4} e^{3} + \frac {9}{4} \, a^{2} b c d^{2} x^{4} e^{3} + \frac {1}{4} \, a^{3} d^{3} x^{4} e^{3} + 3 \, a^{2} b c^{3} f x^{3} e^{2} + 3 \, a^{3} c^{2} d f x^{3} e^{2} + a b^{2} c^{3} x^{3} e^{3} + 3 \, a^{2} b c^{2} d x^{3} e^{3} + a^{3} c d^{2} x^{3} e^{3} + \frac {3}{2} \, a^{3} c^{3} f x^{2} e^{2} + \frac {3}{2} \, a^{2} b c^{3} x^{2} e^{3} + \frac {3}{2} \, a^{3} c^{2} d x^{2} e^{3} + a^{3} c^{3} x e^{3} \]

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, algorithm="giac")

[Out]

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

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maple [B]  time = 0.00, size = 861, normalized size = 2.39 \[ \frac {b^{3} d^{3} f^{3} x^{10}}{10}+\frac {\left (a d f +b c f +b d e \right ) b^{2} d^{2} f^{2} x^{9}}{3}+a^{3} c^{3} e^{3} x +\frac {3 \left (a c f +a d e +b c e \right ) a^{2} c^{2} e^{2} x^{2}}{2}+\frac {\left (\left (a c f +a d e +b c e \right ) b^{2} d^{2} f^{2}+2 \left (a d f +b c f +b d e \right )^{2} b d f +\left (2 \left (a c f +a d e +b c e \right ) b d f +\left (a d f +b c f +b d e \right )^{2}\right ) b d f \right ) x^{8}}{8}+\frac {\left (a \,b^{2} c \,d^{2} e \,f^{2}+2 \left (a c f +a d e +b c e \right ) \left (a d f +b c f +b d e \right ) b d f +\left (2 a b c d e f +2 \left (a c f +a d e +b c e \right ) \left (a d f +b c f +b d e \right )\right ) b d f +\left (a d f +b c f +b d e \right ) \left (2 \left (a c f +a d e +b c e \right ) b d f +\left (a d f +b c f +b d e \right )^{2}\right )\right ) x^{7}}{7}+\frac {\left (2 \left (a d f +b c f +b d e \right ) a b c d e f +\left (2 \left (a d f +b c f +b d e \right ) a c e +\left (a c f +a d e +b c e \right )^{2}\right ) b d f +\left (a c f +a d e +b c e \right ) \left (2 \left (a c f +a d e +b c e \right ) b d f +\left (a d f +b c f +b d e \right )^{2}\right )+\left (a d f +b c f +b d e \right ) \left (2 a b c d e f +2 \left (a c f +a d e +b c e \right ) \left (a d f +b c f +b d e \right )\right )\right ) x^{6}}{6}+\frac {\left (2 \left (a c f +a d e +b c e \right ) a b c d e f +\left (2 \left (a c f +a d e +b c e \right ) b d f +\left (a d f +b c f +b d e \right )^{2}\right ) a c e +\left (a c f +a d e +b c e \right ) \left (2 a b c d e f +2 \left (a c f +a d e +b c e \right ) \left (a d f +b c f +b d e \right )\right )+\left (a d f +b c f +b d e \right ) \left (2 \left (a d f +b c f +b d e \right ) a c e +\left (a c f +a d e +b c e \right )^{2}\right )\right ) x^{5}}{5}+\frac {\left (a^{2} b \,c^{2} d \,e^{2} f +\left (2 a b c d e f +2 \left (a c f +a d e +b c e \right ) \left (a d f +b c f +b d e \right )\right ) a c e +2 \left (a d f +b c f +b d e \right ) \left (a c f +a d e +b c e \right ) a c e +\left (a c f +a d e +b c e \right ) \left (2 \left (a d f +b c f +b d e \right ) a c e +\left (a c f +a d e +b c e \right )^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (a d f +b c f +b d e \right ) a^{2} c^{2} e^{2}+\left (2 \left (a d f +b c f +b d e \right ) a c e +\left (a c f +a d e +b c e \right )^{2}\right ) a c e +2 \left (a c f +a d e +b c e \right )^{2} a c e \right ) x^{3}}{3} \]

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 = 0.68, size = 461, normalized size = 1.28 \[ \frac {1}{10} \, b^{3} d^{3} f^{3} x^{10} + \frac {1}{3} \, {\left (b d e + b c f + a d f\right )} b^{2} d^{2} f^{2} x^{9} + \frac {3}{8} \, {\left (b d e + b c f + a d f\right )}^{2} b d f x^{8} + a^{3} c^{3} e^{3} x + \frac {1}{7} \, {\left (b d e + b c f + a d f\right )}^{3} x^{7} + \frac {1}{4} \, {\left (3 \, b d f x^{4} + 4 \, {\left (b d e + b c f + a d f\right )} x^{3} + 6 \, {\left (b c e + a d e + a c f\right )} x^{2}\right )} a^{2} c^{2} e^{2} + \frac {1}{4} \, {\left (b c e + a d e + a c f\right )}^{3} x^{4} + \frac {1}{70} \, {\left (30 \, b^{2} d^{2} f^{2} x^{7} + 70 \, {\left (b d e + b c f + a d f\right )} b d f x^{6} + 42 \, {\left (b d e + b c f + a d f\right )}^{2} x^{5} + 70 \, {\left (b c e + a d e + a c f\right )}^{2} x^{3} + 21 \, {\left (4 \, b d f x^{5} + 5 \, {\left (b d e + {\left (b c + a d\right )} f\right )} x^{4}\right )} {\left (b c e + a d e + a c f\right )}\right )} a c e + \frac {1}{10} \, {\left (5 \, b d f x^{6} + 6 \, {\left (b d e + {\left (b c + a d\right )} f\right )} x^{5}\right )} {\left (b c e + a d e + a c f\right )}^{2} + \frac {1}{56} \, {\left (21 \, b^{2} d^{2} f^{2} x^{8} + 48 \, {\left (b^{2} d^{2} e f + {\left (b^{2} c d + a b d^{2}\right )} f^{2}\right )} x^{7} + 28 \, {\left (b^{2} d^{2} e^{2} + 2 \, {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} f^{2}\right )} x^{6}\right )} {\left (b c e + a d e + a c f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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, 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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mupad [B]  time = 2.23, size = 787, normalized size = 2.18 \[ x^7\,\left (\frac {a^3\,d^3\,f^3}{7}+\frac {9\,a^2\,b\,c\,d^2\,f^3}{7}+\frac {9\,a^2\,b\,d^3\,e\,f^2}{7}+\frac {9\,a\,b^2\,c^2\,d\,f^3}{7}+\frac {27\,a\,b^2\,c\,d^2\,e\,f^2}{7}+\frac {9\,a\,b^2\,d^3\,e^2\,f}{7}+\frac {b^3\,c^3\,f^3}{7}+\frac {9\,b^3\,c^2\,d\,e\,f^2}{7}+\frac {9\,b^3\,c\,d^2\,e^2\,f}{7}+\frac {b^3\,d^3\,e^3}{7}\right )+x^5\,\left (\frac {3\,a^3\,c^2\,d\,f^3}{5}+\frac {9\,a^3\,c\,d^2\,e\,f^2}{5}+\frac {3\,a^3\,d^3\,e^2\,f}{5}+\frac {3\,a^2\,b\,c^3\,f^3}{5}+\frac {27\,a^2\,b\,c^2\,d\,e\,f^2}{5}+\frac {27\,a^2\,b\,c\,d^2\,e^2\,f}{5}+\frac {3\,a^2\,b\,d^3\,e^3}{5}+\frac {9\,a\,b^2\,c^3\,e\,f^2}{5}+\frac {27\,a\,b^2\,c^2\,d\,e^2\,f}{5}+\frac {9\,a\,b^2\,c\,d^2\,e^3}{5}+\frac {3\,b^3\,c^3\,e^2\,f}{5}+\frac {3\,b^3\,c^2\,d\,e^3}{5}\right )+x^6\,\left (\frac {a^3\,c\,d^2\,f^3}{2}+\frac {a^3\,d^3\,e\,f^2}{2}+\frac {3\,a^2\,b\,c^2\,d\,f^3}{2}+\frac {9\,a^2\,b\,c\,d^2\,e\,f^2}{2}+\frac {3\,a^2\,b\,d^3\,e^2\,f}{2}+\frac {a\,b^2\,c^3\,f^3}{2}+\frac {9\,a\,b^2\,c^2\,d\,e\,f^2}{2}+\frac {9\,a\,b^2\,c\,d^2\,e^2\,f}{2}+\frac {a\,b^2\,d^3\,e^3}{2}+\frac {b^3\,c^3\,e\,f^2}{2}+\frac {3\,b^3\,c^2\,d\,e^2\,f}{2}+\frac {b^3\,c\,d^2\,e^3}{2}\right )+x^4\,\left (\frac {a^3\,c^3\,f^3}{4}+\frac {9\,a^3\,c^2\,d\,e\,f^2}{4}+\frac {9\,a^3\,c\,d^2\,e^2\,f}{4}+\frac {a^3\,d^3\,e^3}{4}+\frac {9\,a^2\,b\,c^3\,e\,f^2}{4}+\frac {27\,a^2\,b\,c^2\,d\,e^2\,f}{4}+\frac {9\,a^2\,b\,c\,d^2\,e^3}{4}+\frac {9\,a\,b^2\,c^3\,e^2\,f}{4}+\frac {9\,a\,b^2\,c^2\,d\,e^3}{4}+\frac {b^3\,c^3\,e^3}{4}\right )+a^3\,c^3\,e^3\,x+\frac {b^3\,d^3\,f^3\,x^{10}}{10}+\frac {3\,a^2\,c^2\,e^2\,x^2\,\left (a\,c\,f+a\,d\,e+b\,c\,e\right )}{2}+\frac {b^2\,d^2\,f^2\,x^9\,\left (a\,d\,f+b\,c\,f+b\,d\,e\right )}{3}+a\,c\,e\,x^3\,\left (a^2\,c^2\,f^2+3\,a^2\,c\,d\,e\,f+a^2\,d^2\,e^2+3\,a\,b\,c^2\,e\,f+3\,a\,b\,c\,d\,e^2+b^2\,c^2\,e^2\right )+\frac {3\,b\,d\,f\,x^8\,\left (a^2\,d^2\,f^2+3\,a\,b\,c\,d\,f^2+3\,a\,b\,d^2\,e\,f+b^2\,c^2\,f^2+3\,b^2\,c\,d\,e\,f+b^2\,d^2\,e^2\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.25, size = 1018, normalized size = 2.82 \[ a^{3} c^{3} e^{3} x + \frac {b^{3} d^{3} f^{3} x^{10}}{10} + x^{9} \left (\frac {a b^{2} d^{3} f^{3}}{3} + \frac {b^{3} c d^{2} f^{3}}{3} + \frac {b^{3} d^{3} e f^{2}}{3}\right ) + x^{8} \left (\frac {3 a^{2} b d^{3} f^{3}}{8} + \frac {9 a b^{2} c d^{2} f^{3}}{8} + \frac {9 a b^{2} d^{3} e f^{2}}{8} + \frac {3 b^{3} c^{2} d f^{3}}{8} + \frac {9 b^{3} c d^{2} e f^{2}}{8} + \frac {3 b^{3} d^{3} e^{2} f}{8}\right ) + x^{7} \left (\frac {a^{3} d^{3} f^{3}}{7} + \frac {9 a^{2} b c d^{2} f^{3}}{7} + \frac {9 a^{2} b d^{3} e f^{2}}{7} + \frac {9 a b^{2} c^{2} d f^{3}}{7} + \frac {27 a b^{2} c d^{2} e f^{2}}{7} + \frac {9 a b^{2} d^{3} e^{2} f}{7} + \frac {b^{3} c^{3} f^{3}}{7} + \frac {9 b^{3} c^{2} d e f^{2}}{7} + \frac {9 b^{3} c d^{2} e^{2} f}{7} + \frac {b^{3} d^{3} e^{3}}{7}\right ) + x^{6} \left (\frac {a^{3} c d^{2} f^{3}}{2} + \frac {a^{3} d^{3} e f^{2}}{2} + \frac {3 a^{2} b c^{2} d f^{3}}{2} + \frac {9 a^{2} b c d^{2} e f^{2}}{2} + \frac {3 a^{2} b d^{3} e^{2} f}{2} + \frac {a b^{2} c^{3} f^{3}}{2} + \frac {9 a b^{2} c^{2} d e f^{2}}{2} + \frac {9 a b^{2} c d^{2} e^{2} f}{2} + \frac {a b^{2} d^{3} e^{3}}{2} + \frac {b^{3} c^{3} e f^{2}}{2} + \frac {3 b^{3} c^{2} d e^{2} f}{2} + \frac {b^{3} c d^{2} e^{3}}{2}\right ) + x^{5} \left (\frac {3 a^{3} c^{2} d f^{3}}{5} + \frac {9 a^{3} c d^{2} e f^{2}}{5} + \frac {3 a^{3} d^{3} e^{2} f}{5} + \frac {3 a^{2} b c^{3} f^{3}}{5} + \frac {27 a^{2} b c^{2} d e f^{2}}{5} + \frac {27 a^{2} b c d^{2} e^{2} f}{5} + \frac {3 a^{2} b d^{3} e^{3}}{5} + \frac {9 a b^{2} c^{3} e f^{2}}{5} + \frac {27 a b^{2} c^{2} d e^{2} f}{5} + \frac {9 a b^{2} c d^{2} e^{3}}{5} + \frac {3 b^{3} c^{3} e^{2} f}{5} + \frac {3 b^{3} c^{2} d e^{3}}{5}\right ) + x^{4} \left (\frac {a^{3} c^{3} f^{3}}{4} + \frac {9 a^{3} c^{2} d e f^{2}}{4} + \frac {9 a^{3} c d^{2} e^{2} f}{4} + \frac {a^{3} d^{3} e^{3}}{4} + \frac {9 a^{2} b c^{3} e f^{2}}{4} + \frac {27 a^{2} b c^{2} d e^{2} f}{4} + \frac {9 a^{2} b c d^{2} e^{3}}{4} + \frac {9 a b^{2} c^{3} e^{2} f}{4} + \frac {9 a b^{2} c^{2} d e^{3}}{4} + \frac {b^{3} c^{3} e^{3}}{4}\right ) + x^{3} \left (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}\right ) + x^{2} \left (\frac {3 a^{3} c^{3} e^{2} f}{2} + \frac {3 a^{3} c^{2} d e^{3}}{2} + \frac {3 a^{2} b c^{3} e^{3}}{2}\right ) \]

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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