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]

((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)

________________________________________________________________________________________

Rubi [A]  time = 0.658437, 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, 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 2059

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

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]))

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*}

Mathematica [A]  time = 0.204866, size = 653, normalized size = 1.81 \[ \frac{3}{8} b d f x^8 \left (a^2 d^2 f^2+3 a b d f (c f+d e)+b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac{1}{7} x^7 \left (9 a^2 b d^2 f^2 (c f+d e)+a^3 d^3 f^3+9 a b^2 d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b^3 \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )\right )+\frac{1}{2} x^6 \left (3 a^2 b d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+a^3 d^2 f^2 (c f+d e)+a b^2 \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )+b^3 c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac{3}{5} x^5 \left (a^2 b \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )+a^3 d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+3 a b^2 c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b^3 c^2 e^2 (c f+d e)\right )+\frac{1}{4} x^4 \left (9 a^2 b c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )+a^3 \left (9 c^2 d e f^2+c^3 f^3+9 c d^2 e^2 f+d^3 e^3\right )+9 a b^2 c^2 e^2 (c f+d e)+b^3 c^3 e^3\right )+a c e x^3 \left (a^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )+3 a b c e (c f+d e)+b^2 c^2 e^2\right )+\frac{3}{2} a^2 c^2 e^2 x^2 (a c f+a d e+b c e)+a^3 c^3 e^3 x+\frac{1}{3} b^2 d^2 f^2 x^9 (a d f+b c f+b d e)+\frac{1}{10} b^3 d^3 f^3 x^{10} \]

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 = 0.001, size = 861, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

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 = 1.20328, size = 622, normalized size = 1.72 \begin{align*} \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 )} \end{align*}

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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Fricas [B]  time = 1.12296, size = 2136, normalized size = 5.92 \begin{align*} \text{result too large to display} \end{align*}

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

________________________________________________________________________________________

Sympy [B]  time = 0.216434, size = 1018, normalized size = 2.82 \begin{align*} \text{result too large to display} \end{align*}

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)

________________________________________________________________________________________

Giac [B]  time = 1.10761, size = 1311, normalized size = 3.63 \begin{align*} \text{result too large to display} \end{align*}

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